I note that the Ellison script had in the changed timeline the name of the ship being the SS Condor: https://memory-beta.fandom.com/wiki/Harlan_Ellison%27s_The_City_on_the_Edge_of_Forever,_Issue_2.
This ship's personnel were not dissimilar to the Mirror universe's ship's (is it still called Enterprise?) and I wonder if it is possible that Ellison's idea of a brutal timeline caused by actions in the past managed to, for example, be brought up in a brain-storming session and this resulted in Mirror, Mirror. Now, I do not find anything about Ellison complaining about this which I would think he would have had he noticed it or not signed away the rights to the ideas in his script.
I think other have noticed some relationship twixt the two stories whether intentional or not: https://www.reddit.com/r/startrek/comments/79cm1j/mirror_universe_theory_and_city_on_the_edge_of/.
One can say, this is just fiction, but so was Shakespeare. Star Trek has become certainly the biggest thing of its kind in history, no other show has inspired so many further ideas by such a diverse group of contributors over such a long period of time. And who knows what the next half century will bring us?