In multiple sources (say here and here) I've see it asserted that a classification of topological spaces up to homeomorphism is either impossible, undesirable because homeomorphism is too strong, or both. (So, the next sentence will go, we should try classifying spaces up to homotopy equivalence instead).
Unfortunately, I've never seen someone elaborate on why classifying spaces by homeomorphism is either impossible or inconveniently strong. Regarding the former statement, what's the easiest way of demonstrating that classification up to homeomorphism is ridiculously hard or impossible, or a particularly good example of why that's the case?
But more importantly, the latter statement - that classification up to homeomorphism wouldn't even be a desirable thing anyway, because it's too fine an equivalence - is not obvious to me at all. In many realistic contexts, homeomorphism isn't even strong enough for what we want to achieve, and we instead need to talk about things like diffeomorphisms between smooth manifolds. Homotopy equivalence is often inconveniently weak, as it doesn't respect topological properties that are pretty important, like compactness in particular. So I don't get why classification up to homotopy equivalence would be a priori better, only why it would be simpler - but as I've said, those two sources seem to me like they are sort of making the claim that it would be better.
So there are some steps missing here for me. I'd appreciate either a quick explanation or, preferred, a reference for a longer one.
Often when people talk about "classifying" they think about having an algorithm (or a method) such that it takes a pair of spaces as an input and returns an answer to the question "are they homeomorphic/homotopic/diffeormorphic, etc?" as an output. All of that in a finite number of steps (see: decidability).
Such algorithm cannot exist because in particular we would be able to restrict the algorithm to manifolds. And in manifold case it is know that the problem is at least as hard as the word problem. And the word problem is known to have no solution.
Also IMO it would be very desirable if possible. I mean, what exactly would be a disadvantage of having such method?