Heuristic argument that finite simple groups _ought_ to be "classifiable"?

It is unlikely that there is any easy reason why a classification is possible, unless someone comes up with a completely new way to classify groups. One problem, as least with the current methods of classification via centralizers of involutions, is that every simple group has to be tested to see if it leads to new simple groups containing it in the centralizer of an involution. For example, when the baby monster was discovered, it had a double cover, which was a potential centralizer of an involution in a larger simple group, which turned out to be the monster. The monster happens to have no double cover so the process stopped there, but without checking every finite simple group there seems no obvious reason why one cannot have an infinite chain of larger and larger sporadic groups, each of which has a double cover that is a centralizer of an involution in the next one. Because of this problem (among others), it was unclear until quite late in the classification whether there would be a finite or infinite number of sporadics.

Any easy way to get around this has been overlooked by about a hundred finite group theorists.

There is a paper of Larsen and Pink (Update: It has appeared: Larsen, Michael; Pink, Richard: Subgroups of algebraic groups. J. Amer. Math. Soc. 24 (2011), 1105–1158.)dating back to 1998 (but still in the process of getting published - long story there) that gives a conceptual proof (based on algebraic geometry methods, primarily) that any sufficiently large finite simple group that has a bounded rank linear model (i.e. it is isomorphic to a subgroup of $GL_d(k)$ for some field k and some bounded d) is basically of Lie type. So this, combined with the classification of simple groups of Lie type, gives an answer to the question in the bounded rank case. Unfortunately this isn't the whole story because one can certainly let the rank go to infinity, and then there are also the pesky alternating groups which are not of Lie type at all (except, perhaps, over the field of one element, whatever that means...).

I can't tell you much about finite groups, but I can tell you that unfortunately there is no general model-theoretic result along the lines of "If a collection of objects has a simple axiomatization, then it must be easy to classify." In fact, considering some examples, I believe that no such general result could exist even in principle.

Finitely axiomatizable, but hard to classify: The class of all linearly-ordered sets. There are many ways to make precise the idea that these are "hard to classify:" for any uncountable cardinal $\kappa$, there are many (i.e. $2^\kappa$) pairwise nonisomorphic orderings of size $\kappa$; there is no way to characterize arbitrary linear orderings up to isomorphism by a fixed set of cardinal-number invariants; and there are many large families of linear orderings which "look similar" but are nonisomorphic (where "looks similar" could be mean various things: bi-embeddable by maps preserving the truth of all first-order formulas, or "there is a forcing extension of the universe of set theory that preserves all cardinal numbers, adds no new subsets of $\mathbb{R}$, and in which the two structures are isomorphic," etc...)

Easy to classify, but not finitely axiomatizable: For example, the set of all algebraically-closed fields. These are axiomatizable by an infinite list of axioms: take all the field axioms, plus, for each natural number $n > 0$, an axiom saying "every degree-$n$ polynomial has at least one root." However, a simple argument using the compactness theorem shows that this class cannot be finitely axiomatizable. Also, these structures are "very easy to classify" in the sense that they are characterized, up to isomorphism, by just two cardinal numbers: the characteristic and the transcendence degree (over the prime subfield). (And hence any two such fields that ``look similar'' in the senses I mentioned above must actually be isomorphic.)

In fact, it's worse than these examples suggest. There is a theorem in model theory due to Cherlin, Harrington, and Lachlan saying that any axiomatizable class that is ``easy to classify'' in the sense that there is just one member (up to isomorphism) of size $\kappa$ for any infinite cardinal $\kappa$ cannot be finitely axiomatizable!

There is a well-studied notion of "classifiability" in model theory which concerns how hard it is to characterize all the models of a given theory by a "reasonable set of invariants." The main reference is Shelah's monograph Classification Theory. In general, classifiability of a theory has no logical relation with how hard it is to axiomatize the theory (e.g. whether it is finitely axiomatizable, computably axiomatizable, etc.). But Shelah's classification theory only treats theories with only infinite models, so I'm not sure that it can answer your question about finite simple groups.