I don't know exactly where you came across the idea that 'knowledge (viz. the ability to know that) is invariably linked to propositional modal logic', but there are a couple of things, both of which are worth giving a brief overview of. These are epistemic logic -- the logic governing statements of ascriptions of knowledge -- which is a form of modal logic, and knowability -- questions about when it is possible to know something. This is naturally formalised in a propositional modal logic (perhaps a modal epistemic logic).
We might ask -- what is the appropriate logic for reasoning about knowledge? If I make the following argument:
- I know that it is sunny outside
- I know that if it is sunny outside then there is less than 100% cloud cover
- Therefore: I know that there is less than 100% cloud cover.
Is this argument valid? If so, why? What are the principles that govern knowledge ascriptions.
Here's how we might start: Suppose that we start with propositional logic -- the logic of 'and', 'or', 'if-then', 'not' and so on. How do we add knowledge statements? It is natural to think that knowledge is knowledge of a proposition. So, we add a symbol 'K', so that for any sentence φ, 'Kφ' is well-formed, intuitively read as 'It is known that p'. (We might subscript K with a letter denoting who knows that p'.
What principles should we expect. Here are a couple:
(Nec) If you can derive φ without assumptions, then derive Kφ.
(K) Kφ & K(φ→ψ) → K(ψ)
The first of these says that, if we've proved that φ, then φ is known. This is a bit controversial, since it seems to suggest that all logical truths are known. But perhaps we can brush this aside and assume we are dealing with some kind of idealised perfectly rational being.
The second says that knowledge is preserved under known entailment. Seems pretty plausible to me.
Anyway, if K satisfies these principles, it means that the logic is what is called a normal modal logic. So that's one way to think that modal logic might be important for epistemology.
You can read more about it at the Stanford Encyclopedia of Philosophy here
I won't say much about this. But there's been a lot of discussion about knowability. So, for example, we might as whether every truth is knowable, in principle. Although this seems quite plausible, it turns out that, with some plausible assumptions, it entails that every truth is known. This is known as the knowability paradox.
This topic is linked to modal logic since it's about the possibility of knowledge. And lots of writers on the issue make use of modal logic when discussing it.
You can read more about this, again at the SEP, here
(You mention intuitionistic attitudes to knowledge. This topic is very closely related to that, in that people have tried to use the knowability principle -- that every truth is knowable -- to argue for intuitionistic logic. So it might be something you'd be interested in reading up on.)
Does this matter?
I'm conscious that I've not really answered your question. I've instead just tried to give a few details about some of the topics where issues to do with knowledge are tied up with issues to do with modal logic.
But must it be tied up in this way? No, absolutely not. There's plenty of discussion in epistemology which has nothing to do with modal logic. Again, SEP is a good guide to some of the issues that are popularly discussed: Epistemology at SEP. Many of these discussions are explicitly concerned with your last question: 'do philosophers really know what knowledge is?'. The answer: no, but they're working on it!