Not even Superman can save everyone. I think this only humanizes him even more, this probably weighs on Clark's thoughts everyday. Even he knows that he can't save everyone even as powerful as he is. I wouldn't call it an "inherent problem"...I'd call it part of his human upbringing and perhaps even an endearing character trait. Despite all that tremendous power he still can't save everyone.
We see a partial example of this in SR. I'm speaking of the scene where he hovers above Earth listening for cries for help with his super hearing, and then zooms down to the rescue after apparentely deciding on who to save. In a real world setting, Superman wouldn't be like
a god, he would become "god".
I imagine that say, someone is going down in a plane; knowing Superman's super hearing, those people would cry out to Superman instead of God. Because, it may be in God's "plan" that they die, but Superman (if you were the chosen ones) would save you regardless. It wouldn't be long before people would fully deify Supes and start worshipping his iconic image.
But the pressure to save everyone would mount. "Superman, why would you save them and not us, we have suffered so". "Superman, you always save them and never save us, why?".
He might end up sorry Jor-El didn't just keep him on Krypton.