Ok, not "anybody" as in a "normal guy" as you say, but the role wasn't initially written as Khan, and that's enough to know that it wasn't necessary for it to be Khan. Adding in a bunch of stuff like the Worf Effect was just superficial trimmings to fit Khan, not elements that were pivotal to the story. Were it someone else, the story would've accommodated them. This isn't that hard to see.
So basically instead of Khan, you want yet another Khan ripoff?
Nothing in what Cumberbatch does in STD reminds me of Khan, so I never get that argument. When was Khan ever able to beat a bunch of Klingons to a pulp, shrug off phaser blasts, resist Vulcan nerve pinches and have healing blood? Seriously, nobody would even THINK about shouting "Khan Ripoff!" if the guy was just called "John Harrison, enhanced secret agent".
Make Harrison a human-alien hybrid to explain his abilities and there's NOTHING left of Khan.