Yes, if you apply hard logic to comics, all of it will fall apart. But you can ask the reader to accept that some people are born with a gene that gives them special powers, or that a radioactive spider bite will transfer a spider's abilities. That's all in the realm of pure fantasy and if you don't buy into that, you're not interested in the story anyhow. But within that fantasy, there should be a certain sense of plausibility to help ground what's happening in an otherwise recognizable reality.
Having a character be able to believably maintain a secret identity is just harder now than it used to be. For some character's it's worth keeping that aspect up. For a lot of others, it really doesn't matter all that much.
If I had to pick one character to restore their secret identity, it'd be Captain America. I always liked it when he had his supporting cast who didn't know who he was and that he had to make a living as a freelance artist. On the other hand, the idea that Captain America has to hold down a regular job to pay the bills on top of his responsibilities to the nation seems like an unnecessary inconvenience. I mean, if being Captain America isn't a full time job in and of itself, what is?