Wally worked for years with a public identity until Geoff Johns first
Flash run, I believe. Heroes can have public identities, but other writers just fall back on "your loved ones will be in trouble" trope. Which reminds me, I need to finish Mark Baron's run. As for Superman, John Byrne effectively, until it was retconned out, stripped Superman of his Kryptonian heritage with the birthing Matrix. Clark's "real" parents was the Kents. That's what he was, a human. An American human at that (American human, was that some phrase back in 1987?). So essentially, there was no otherness with Superman. But, later writers didn't care about that and wrote him as the alien he is.
When it comes to the "Superman's humanity" argument, we all have our own idealized Superman and how he should act. Some folks wants the "he's the most human of us all", others want the pre-Crisis Supergod, or Superdad, even some with the "edgy and dark" New52. So, in the end, it's a moot discussion because everyone won't come to an agreement on the character.