Is the following intuition valid for understanding $k$-wise independent hash functions?
Your intuition is exactly right.  Yes, that's equivalent to choosing a random polynomial over $\mathbb{F}_p$.  The reason why it works is exactly the interpolation theorem for finite fields.

$k$-wise independent basically means "it behaves like a perfect hash function, if you only feed it $k$ inputs" or "it behaves like a perfect hash function, as far as correlations between up to $k$ different hash values appear"; and the interpolation theorem then ensures that this level of perfection is achieved for the construction you outline.

It can be a bit tricky to explain this intuition to students who aren't familiar with anything from number theory, group theory, or finite fields.  One approach is to explain the intuition of the interpolation theorem over the real numbers, and then just give a statement that "something similar happens when you work modulo $p$ a prime", without trying to prove it.  That's not perfect, though.

Another approach is to not try to teach this topic, at this point in their education.  While I was taught universal hashing in my undergraduate algorithms course, I've gradually come to the opinion that the topic is better to omit from the course: it's not important enough, and it relies too much on mathematical background and intuition that students don't have.  Why isn't it important?  Well, most students won't become theorists.  For students who will eventually become practitioners, they probably won't ever need universal hash functions.  Instead, ordinary hash functions are often so good that as a first approximation you can just model them as if they were universal (as a heuristic), without making a big deal about it, and do your probability calculations from there.  This is enough to let you analyze uses of hash functions in a way that's accurate enough for practical/engineering purposes.  And, given that, it's not clear that we need to show them constructions that provably achieve something that ordinary hash functions already heuristically achieve pretty well.  That's just a personal opinion, though.