The "derived drift" is pretty unsatisfying and dangerous to category theory (or at least, to me)
Higher category theory is, roughly speaking, where category theory meets homotopy coherent mathematics. It is hence relevant to those problems in which categorical structures and homotopy coherent phenomena play a significant role. Many areas of algebraic topology and algebraic geometry have this property. There are also many such areas who don't. From what I understood from your question, you like category theory, but not so much homotopy coherent mathematics. So far I would say you don't actually have any problem, since ordinary category theory itself is not, at least in my opinion, a domain in which homotopy coherent mathematics is crucially needed. This is mainly because the coherence issues that arise in category theory are very low-dimensional, to the extent that it is more cost effective to do them by hand (or simply neglect them), then to use fancy machinery. This leads me to the first possible solution to your problem:
Do category theory.
It has really not been my impression that this field is anywhere close to finished. This is especially true if you consider 2-category theory as an acceptable extension (here the coherence is again usually simple enough to do by hand). It also has many interactions with domains such as logic, set theory and foundations of mathematics. You will find many interesting discussions of all these topics, as well as links to state-of-the-art research, in the n-category café. It will also not surprise me if you will find people there who share your mathematical taste.
If you still maintain, for whatever reason, that it is imperative that you do things related to higher category theory, I can tell you that there are many domains in this topic which are very 1-categorical in flavor. For example, you can
Do model category theory.
This notion, one of many brilliant ideas of Quillen, allows one to magically reduce homotopy coherence issues into a 1-categorical framework. Model categories also share many of the aesthetic features of ordinary category theory, in the sense that everything seems to fit together very nicely, while still being extremely useful for real world homotopy coherent mathematics. A bit less known, but also very categorical in flavor are derivators. You might also look into triangulated categories.
Finally, as many of the comments above suggest, it's possible that the things that you don't like in homotopy coherent mathematics are actually not essential properties of the field, but rather of its young age. You may hence consider to
Give HTT another try.
In doing so, you may want to take into account the following: I strongly believe that no one has ever written a technical simplex-by-simplex combinatorial proof of an HTT-type result without knowing in advance that what they want to prove is true, and moreover why it is true. This is because, despite the technicality of some proofs, higher categories do behave according to fundamental principles. Sometimes these principles are the same as the 1-categorical case, but sometimes they're different. As a result, it may take a bit of time to acquire a guiding intuition for what should be true and when. It is, nonetheless, certainly doable. I would then suggest that, before reading a given proof, you try to think first why the announced result should be true. In addition, think how you would prove, say, the 1-categorical case, and then try to extend the proof to higher categories dimension by dimension, and see where this leads you. Then read the simplex-by-simplex argument. It may suddenly look very clear.
This is too long for a comment, but doesn't exactly answer the question. However, I've had enough eggnog this Christmas that I'm going to post it anyway (despite knowing almost nothing about category theory).
Reading the question and skimming over the comments, I see a lot of romantic descriptions of the practice of mathematics that bear little relationship to how it is actually practiced. It is truly wonderful when a single elegant idea can completely illuminate and render transparent some part of the subject. However, these ideas are usually the end product of a long development that starts with a hacked together, complicated mess of arguments. And they are discovered by people who are deeply immeshed in the subject.
To put it another way, while it is great to have a strong philosophical take on what mathematics is and how it should be practiced, if that philosophy is not informed by the actual practice of mathematics, then it is unlikely to lead anywhere. Philosophical clarity comes at the end and not at the beginning.
To be successful at research, you have to be willing to get your hands dirty. If you don't enjoy the ordinary craft of doing mathematics, then it is unlikely that you will be happy as a research mathematician. But it is a craft. I strongly disagree with various comments that make it sound like you have to be some kind of crazy romantic hero taking superhuman risks or something. I certainly am not like that, but I have been able to make a career out this.
Now, it is impossible for us to give you personal advice on what you should do with your life or what direction your research should take. We don't know you. But I can say that everyone goes through periods of doubt and frustration. What I always do in those situations is to take a brief break from the front lines of research and go back to the sources that drew me to mathematics in the first place. Read some great mathematics, be refreshed, and then get back at it.
I'm coming a bit late to this party, but I'll put in my two cents anyway because they are rather different from everything else I've heard so far. In a nutshell, my response is:
Yes, I agree that this is a problem (though I do think you would have done better to post only the question and not the rant), and
What you can do is be part of the solution.
For a long time I resisted "homotopical higher category theory" too, for reasons that I think are not unrelated to yours. I even wrote a somewhat whiny blog post about it. What eventually "brought me on board" was not the applications to algebraic geometry or what-have-you (which is not, of course, to denigrate those applications), but the truly category-theoretic conceptual insights arising from what you call HTT. Examples include:
Colimits in a 1-category cannot be as well-behaved as we would like them to be, and the reason is because a 1-category doesn't have enough "room"; an $(\infty,1)$-category fixes this. For instance, Giraud's axioms for a 1-topos assert "descent" only for coproducts and quotients of equivalence relations; the analogous axioms for an $(\infty,1)$-topos assert descent for all colimits.
Passing "all the way to $\infty$" has a "stabilizing" effect that enables $(\infty,1)$-categories and $(\infty,1)$-category theory to "describe itself" in ways that 1-category theory can only approximate. For instance, the 1-category of 1-categories does not include enough information to characterize the "correct" notion of "sameness" for 1-categories, namely equivalence (at least, not unless you hack it with something like a Quillen model structure); for that you need the 2-category of 1-categories, or at least the $(2,1)$-category of 1-categories. But the $(\infty,1)$-category of $(\infty,1)$-categories does characterize them up to the correct notion of equivalence. (Although for many purposes one still needs the $(\infty,2)$-category of $(\infty,1)$-categories, pointing towards the still largely-unexplored territory of $(\infty,\infty)$-categories.) Similarly, a 1-topos can only have a subobject classifier, classifying those objects that are "internally $(-1)$-categories", i.e. truth values; but an $(\infty,1)$-topos can have an object classifier that classifies all objects (up to size limitations).
Various mysterious phenomena in 1-topos theory are explained as shadows of $(\infty,1)$-topos-theoretic phenomena. For instance, the analogy between open geometric morphisms and locally connected ones is explained by seeing them as the steps $k=-1$ and $k=0$ of a ladder of locally $k$-connected $(\infty,1)$-geometric morphisms, and similarly for proper and tidy geometric morphisms. Moreover, various apparently ad hoc notions of the "homotopy theory of toposes", such as cohomology, fundamental groups, shape theory, and so on, are explained as manifestations of the $(\infty,1)$-topos-theoretic "shape", which is characterized by a simple universal property.
Perhaps most importantly, the fundamental idea that the basic objects of mathematics are not just sets, but $\infty$-groupoids. Thus, for instance, the really good notion of "ring" should be an $\infty$-groupoid with a coherent multiplication and addition structure (i.e. a ring spectrum), including the set-based notion of "ring" as simply a special case. And so on.
Note that none of these ideas depends on any concrete model for $(\infty,1)$-categories, and most of them have nothing to do with homotopy theory; they are purely category-theoretic ideas. So I think even a category theorist who cares nothing about homotopy theory ought to be interested in a kind of "category theory" where these are true.
That said, I think a good category theorist should care at least somewhat about homotopy theory, if for no other reason then for the same reason that a good category theorist should care about other applications of category theory. Like all fields of mathematics, category theory is supported and invigorated by its connections to other fields of mathematics, and the close tie between higher category theory and homotopy theory has great potential to stimulate both subjects. That this potential has been realized more fully on the homotopy-theoretic side is, I think, largely an accident of history and personality.
Why is $(\infty,1)$-category theory not usually done "Australian-style"? I believe it is just because people doing $(\infty,1)$-category theory don't know, or at least don't appreciate, Australian-style 1- and 2-category theory, while many Australian-style category theorists don't know or appreciate $(\infty,1)$-category theory. This creates a tremendous opportunity for anyone who is willing to put in the effort to be a bridge, teaching category theorists how to think about $(\infty,1)$-categories "category-theoretically" and teaching $(\infty,1)$-category theorists the benefits of "really thinking like a category theorist".
One way to be such a bridge is to learn the simplicial technology that's currently used for $(\infty,1)$-category theory and "do them Australian-style". For instance, as far as I know there is still no $(\infty,2)$-monad theory with the power and flexibility of 2-monad theory; someone should do it. Enriched $(\infty,1)$-categories are only starting to be investigated. The $(\infty,2)$-category of $(\infty,1)$-profunctors has been used for some applications, but its category-theoretic potential is largely unexplored. As far as I know, no one has even defined $\infty$-double-categories yet. (Edit: They've been defined, but apparently not systematically studied; see comments.) What about generalized $\infty$-multicategories? Etc. etc.
While a worthy endeavor, I suspect that this is not what you want to do. In particular, it sounds like you don't feel able to spend the time to really understand simplicial technology. I can sympathize with that; it's difficult enough for me, and I was already exposed to lots of simplicial stuff as a graduate student since my advisor was an algebraic topologist. So I generally avoid using simplicial technology as much as possible. One way to do this, which I have pursued myself, is to study $(\infty,1)$-categories using 1- and 2-categorical machinery, including Quillen model categories (which, by the way, have an algebraic version that is rather more pleasing to a category theorist's heart) but also homotopy-level structures such as derivators, homotopy 2-categories, and homotopy proarrow equipments.
This works quite well for surprisingly many things, and doesn't require you to learn any simplicial technology. However, it does often depend on the fact that someone has proven something using simplicial technology in order to "get into the world" where you're working. Moreover, you've also expressed some skepticism about the very idea of simplicial technology and concrete models. I think it'd be good if you can get over this to a degree — mathematics has to move forward with what we have, even if it's not perfect, and later on someone can make it better — but I do also sympathize with it, because for instance of the last conceptual insight I mentioned above:
- The basic objects of mathematics are not just sets, but $\infty$-groupoids.
How can this be, if an $\infty$-groupoid is defined in terms of sets (e.g. as a Kan complex)?
Well... there is now a way to study $\infty$-groupoids directly, without defining them in terms of sets: it's called homotopy type theory (HoTT). HoTT is (among other things) a foundational theory, on roughly the same ontological level as ZFC, whose basic objects can be regarded as $\infty$-groupoids; I wrote a philosophical introduction to it from this perspective. (There's also work on an analogous theory whose basic objects are $(\infty,1)$-categories.) Thus, HoTT offers the promise of an approach to homotopy theory and higher category theory that's almost completely free of simplicial technology, and incorporates the conceptual insights of $(\infty,1)$-category theory "from the ground up", allowing us to build intuition for, and work directly with, higher-categorical and higher-homotopical structures without having to construct them explicitly out of sets. When I read or write a proof in $(\infty,1)$-topos-theoretic language, I'm never quite sure whether I've dotted enough "i"s to make all the coherence come out right; but when I instead write it in HoTT then I am, not only because with HoTT I understand the profound reason why you already know what things intimately are (as you put it), but because a HoTT proof can be formalized and verified with a computer proof assistant. There are already some graduate students who have "grown up" with HoTT and can "think in it" in ways that surpass those of us who "came to it late".
Now, this "promise" of HoTT is not yet fully realized. Many coherent higher-categorical structures can be represented simply and conceptually in HoTT; but many others we don't know how to deal with yet. So here's another way you can be part of the solution: improve the ability of HoTT to represent higher category theory, so that eventually it becomes powerful enough that even the "applied" $(\infty,1)$-category theorists can do away with simplices. This is, in large part, what I am now working on myself.