Yes, this is something I've always thought about whenever the X-Men came up in relation to other heroes.The only ones I think get a pass are the Fantastic Four, since they were prominent in certain ways before getting powers, and I'm pretty sure their story of being hit by cosmic rays is well known.

How come people don't call Spider-Man a mutant? Have people heard his story of being bitten by a radioactive spider, and do they actually believe it? If they do, why?

It would have been an awesome idea if mutant genes were the whole reason for people having powers in the Marvel Universe that would give it a great overarching significance and get rid of these kinds of plot holes.

Of course we can't really expect that much. The Marvel "Universe" was assembled higgledy-piggledy from various comics that happened to be running at around the same time and only years after the fact was the concept of continuity and internal consistency taken seriously.