Look, if you want to speculate as to how different the film might be if Khan wasn't Khan, that's all well and good, but that's not what the original premise of this thread seemed to be. My understanding of the original premise was "don't have Khan be Khan, but otherwise minimal changes to the film", in which case I think my concern is valid.
Whether or not it's pertinent to the film that Khan is from Earth's past, it is brought up in the film. If we're going by the theory of minimal changes, then some generic character from Earth's past being in the film might make me wonder why they didn't use a classic villain. If we're -not- going by that theory, then you can't just say "he's not Khan", but rather you've opened up the question of what his background is instead and how it's significantly different from Khan's.