Khan acknowledging Chekov without ever knowing the man
... is an easy fix that you're rather desperately trying to inflate into a problem of plot logic. TOS adventures were not particularly linear in in-setting timeline, nor is it ever definitively stated in-universe when Chekov was on Enterprise. It's perfectly possible and breaks nothing in the original episode story to simply postulate that Chekov was on board without being on screen (as indeed a great deal of the action in any episode take place offscreen).
It's just not a comparable problem to "Starfleet suddenly promotes cadets on academic probation to commanding flagships." A fix can be found there, too; it's just mostly, when it comes right down to it -- this problem has been extensively chewed over and many defenses attempted in other threads -- along the lines of "forget about the writing and concentrate on the action." Since a fairly major in-universe detail like that (and many other plot logic problems in the Abrams films) just doesn't have a simple, plausible off-screen fix like the supposed Chekov problem.
Sorry, but however much you want them to be, they're not similar cases.