Words like "important" and "natural" are obviously subjective, and that's OK! It shouldn't surprise you to learn that other people have opinions about the value of certain mathematical ideas, just as you probably hope that they aren't surprised that you have opinions, too.

That said, your question suggests a comparison between two types of value judgements: "important" or "natural" vs. "fun" or "curious". The first type suggests that people should work on the problem out of some sense of obligation, whereas the latter suggests that it would be pleasurable to do so. It sounds like you're asking: where does this sense of obligation come from?

The answer is once again obvious: it's determined informally by how the mathematical community allocates resources - journal space, grant money, academic jobs, and so forth. The community has certain informal standards, expressed very beautifully by Gil Kalai's list, for instance.

But presumably those standards exist to ensure that the mathematical community is achieving some sort of larger objective, which makes it worthy of investing resources in the first place. I think that objective is something like "organize and preserve mathematical knowledge". As more and more mathematical knowledge is generated, it becomes harder for even a modestly sized group of experts to keep track of it all. So in order for all of that knowledge to survive in a form that can be easily consumed and appropriated by future generations, there has to be a constant process of simplifying and clarifying the most fundamental ideas.

So that's what I think "important" means. A student today can graduate college knowing how to solve a dozen problems each of which past mathematicians spent their entire lives working on, and it's because the intervening generations isolated a small number of crucial ideas that tie it all together - things like the definition of a limit, or Fourier series, or Galois theory. I think many mathematicians today feel an obligation to provide a similar service for students of the future.