Why doesn't the independence of the continuum hypothesis immediately imply that ZFC is unsatisfactory?

It is true that, if ZFC is consistent, CH is undecidable within it. This is just one example of a more general fact: any consistent recursively enumerable first-order theory at least as strong as Peano arithmetic contains a statement undecidable in that theory. This is the first of Gödel's incompleteness theorems; the second gives an example, namely a statement calling the theory consistent.

The real question isn't whether ZFC has known limitations of this kind; of course it does. The question is which other statements we should add as axioms. The C in ZFC is the axiom of choice, which is itself undecidable in ZF. The history of set theory has seen much broader support in favour of adding AC than in favour of subsequently adding CH.

Why? Well, let's look at some of the differences:

• Although AC was initially much more controversial than it is today, it has come to enjoy broad support for the simple reason that, although it has some counter-intuitive consequences such as Zermelo's well-ordering theorem, its negation has "even worse" consequences such as trichotomy violation.
• Not only is $\beth_1=\aleph_1$ (i.e. the CH) undecidable in ZFC; so is $\beth_1=\aleph_n$ for any positive integer $n$. Why would we adopt the first as an axiom? By contrast, AC doesn't have an infinite family of obvious counterparts that feel equally feasible.
• One good thing about the CH is that it's a special case of a more general idea that looks nice, the GCH ($\beth_\alpha=\aleph_\alpha$ for all ordinals $\alpha$). There is some interest in adding GCH to ZFC, but just CH on its own? That's an unpopular compromise between ZFC (which is weak enough for some people's tastes) and ZFC+GCH (which is strong enough for some other people's tastes).
• Lastly, outside of set theory $\aleph_1$ usually doesn't even come up as a concept, and so CH has no obvious benefit when founding any other areas of mathematics. (There are exceptions, e.g. nonstandard analysis uses CH to consider real infinitesimals.) By contrast, AC has uses all over the place, e.g. proving every vector space has a basis.

There are two issues that I see with your question.

The first and foremost is using the terms "true" and "false" without fully understanding them (granted, many mathematicians do this as well).¹ We have a good idea what is $\Bbb N$ and what is $\Bbb R$. So it is easy to talk about truth in analysis as relative to $\Bbb R$ in some rich enough language that encapsulates the real numbers as we understand them; and truth in number theory is just truth in $\Bbb N$ as a model of PA.

Set theory, however, is far less intuitive. This can be easily seen by the many counter-intuitive results in set theory (from Banach–Tarski to the Division Paradox) that follow from the fact that our intuition is simply not very good with infinite objects that have little structure.

So as we do not have a clear and uniform intuition as to what are sets like, not in the same way we have about the natural numbers or even the real numbers, it is hard—even, impossible—to have a canonical model of set theory where we can evaluate every statement and decide whether or not it is true.²

As we have no canonical way to determine truth and false, there is no canonical way to decide if the Continuum Hypothesis, or many many other statements are "true" or "false". And this opens the door to approaches other than Platonism.

For a Platonist, every statement should be either true or false. That is also a naive approach to mathematics. Sure. It's like that in real life, to some extent. Either you are a doctor, or you're not. Either the sun will engulf the earth as it expands to a red giant, or it won't (for one of many reasons). Things in our life tend to have certain absoluteness to them,³ so we try to apply these principles to mathematics as well.

But mathematics has nothing to do with our reality. Especially set theory, or anything which deals with the infinite, really. So why should there be a canonical universe for truth and false? If anything, set theoretic research shows that one the plethora of mathematical universes with very different theories holding true in them, can all be very interesting.

Okay. So there is no true and false here. What about the second issue?

 

The second issue is that it is actually a good thing that ZFC is not a complete theory. It is a good thing that our foundation doesn't tell us all the answers.

Foundational theories are not there to "give us all the answers", they are there "to formalize our arguments into a mathematical context". ZFC does that magnificently for the most part.

There is a lot of research into "removing unneeded hypothesis" from mathematics. You prove something under assumption that a function is analytic. But maybe proving it under smoothness is enough? Maybe just continuous? Maybe just measurable? Maybe any function?

You want foundations that are strong enough to support your work, but not too strong that they make a lot of unneeded assumptions for you. And while I agree, axioms like $V=L$ are sexy, and they resolve a lot of questions (e.g. CH and things like the Suslin Hypothesis), for the most part, mathematics works just peachy without them.

Not only that, once you start putting in set theoretic "decisions" into your foundations, your axiomatization of the concept of "set" will invariably have to become technical. ZFC is simple, it is elegant. Putting CH into it will involve long and technical statements about functions, about cardinals, about so much more. This will muddy the waters, and for what? Yes, CH has consequences in analysis and in general, but are those enough to get CH "canonical"?

The answer seems negative. Not because set theorists don't care, but because there have been little to no pressure from the "working mathematician" side regarding adding new axioms to set theory. And again, this is not without reason.

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Footnotes.

1. Truth is always relative to a fixed structure, in a lot of cases we just have some tacit agreement about the structure.

2. Even in the case of $\Bbb N$ not everyone would agree. For example, "Supercompact cardinals are consistent with ZFC" is a statement about the natural numbers. Some mathematician would say it is true, others would disagree.

3. They really don't, though.

There are classical arguments, which have historically been seen as compelling, which are intended to argue that the ZFC axioms are all accurate statements about the intuitive concept of a pure, well-founded set.

In principle, it would be possible for us to take some new axioms, in addition to ZFC. The addition of these new axioms could, in principle, allow us to prove statements such as CH. In particular, of course, we could just take CH itself as an axiom, in addition to ZFC.

The issue is that, for systems like ZFC, many people like to have a justification for why the axioms should be assumed. We can take any statement as an axiom, just for the sake of reasoning, but for specific systems such as Peano arithmetic, Euclidean geometry, or ZFC, we like to see a "reason" why each axiom is accepted.

Looking at CH in particular, it seems hard to justify why we would take CH (or its negation) as a new axiom of ZFC. Doing so feels like missing the point - it feels like we are arbitrarily taking sides. And some other, "more justified" axiom might come along later but go in the opposite direction from the arbitrary choice we made.

There has been a significant amount of published discussion on this question, which is more philosophical than mathematical. There are some people working in set theory who do feel that new axioms might be discovered that would resolve CH. If a sufficient justification for these axioms could be given, perhaps a large number of mathematicians would view this as resolving the issue of CH. But others in set theory are more pessimistic. One argument they give is that, because we understand so well the way in which CH is independent from ZFC, if some new axiom were to decide the CH question, we could apply the techniques developed for CH to the new axiom, which might cast doubt on accepting the new axiom on equal footing with the rest of ZFC.

Finally, there is a linguistic triviality: some people misuse the word "true" to mean "provable", so if they say "In ZFC, CH is neither true nor false" they only mean "In ZFC, CH is neither provable nor disprovable". In any particular model of ZFC, of course, either CH is true or CH is false. So any claim that "CH is neither true nor false" has to be read in so other way besides talking about truth in some particular model.