The opening credits of Enterprise shows the Terran Empire landing on the Moon after winning what would be WW2 for us. I think the point of divergence is well before Zephram and even the 20th Century.
I think it was depicting the Empire retaking the moon after WWIII (and then fighting over who did the retaking), not a replacement for the original moon landing.
What happened in A Mirror Darkly shows the divergent point from the primary Star Trek timeline, which diverged from our own timeline in the 1960s or 1970s.
Everything in the Mirror timeline makes it highly unlikely that the same people would exist as in the Prime timeline. There's just something about that universe that makes such extreme improbabilities happen regardless.
I think it's explained as the Mirror Universe somehow being specially connected to the Prime one. But yeah, it's really just an excuse to show known characters being assholes.