The “problem” with Clark being the “real” identity is that if Clark is the “real” identity, that means he needs to be brave and tough and driven because that’s who Superman is at his core. But that also means that there’s virtually no difference between the two identities anymore, which as others have pointed out, was basically how George Reeves played it. That’s a perfectly valid take to be fair. As for why some of us might prefer otherwise, here’s a good summary from Dispenser’s tumblr:
https://davidmann95.tumblr.com/post/...the-old-debate
However the best take and the one I personally prescribe to for both Supes and Bats is that both are “real” in that both identities express different aspects of the core person Clark and Bruce are.