Another question is: how thorough a divorce? We saw McCoy, Spock, Scotty, Sarek, and the Bozeman, all of which did not appear different enough to definitively come from an alternate univers. Kor, Kang, and Koloth, arguably could be different universe versions. Dax's backstory comes from TOS.
So, postulating that a divorce occurred, explain these people. Also, even if we assumed TOS does not lead to TNG after all, does this necessarily mean that we would not still have the Betazoids, Bajorans, Cardassians, Breen, Dominion, Q, et alia in the same universe? ST:V, IIRC, originally had a scene with Chekov losing at Chess to a Betazoid. TNG never nailed down, AFAIK, just when we met some of these races / empires.
Does this mean Kirk would never have heard of Aldea or Iconia?