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.
They do this weird thing in many Anime canons where the same original movie/series can have multiple sequels that are themselves set in totally alternate universes and have nothing whatsoever to do with each other. Mostly these are for logistical reasons -- a new creative team gets a contract to do the sequel, which bombs horribly and the studio goes back to the original team -- but it does create some interesting "split canon" situations. The most famous is the Macross series: there's the Robotech version with its sanitized/Americanized continuity, the "Macross-II" version which takes place 80 years after the movie, and then the "Macross Prime" universe that includes Flashback 2012, Macross 7 and Macross Frontier. The original SDF-1 and various characters make appearances and/or cameos in all three subsequent universes, but only the prime one is really "official" as such.
Star Trek has basically started to do this with the Abrams trek version, creating a completely alternate timeline that is similar to but not entirely the same as TOS. We could probably do the same thing with TNG+ and assume that it too is an alternate timeline that is similar to but not entirely the same as TOS' immediate future.
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?
For that matter, the Xindi, the Xyrillians, the Suliban, the Tandarians, the Andorians and the other ENT races we never saw again in later centuries. It stands to reason that different races have different levels of prominence decade after decade; actually, that's sort of the case in the REAL world, if you think about it. In the first half of the 20th century, Germany and the UK were both major players in two consecutive world wars; now their international profile is secondary at best, and Germany hasn't participated in a major war since 1945. OTOH, alot of the "household names" in international affairs didn't even exist 60 years ago (Iran and North Korea, for example) or if they did, were isolated backwaters nobody had ever heard of or cared about.
I'm thinking that if we followed the TOS timeline as it progressed from the movies, we would see the Klingons become more peaceful partners in the Federation by the start of the 24th century, followed by the Romulans shortly thereafter. Those three powers would probably become helpful allies to Starfleet and the other Federation members against other threats -- the Cardassians and the Breen or whoever else might show up in the new timeline -- and their subsequent disputes would never amount to more than diplomatic/economic squabbles over exchange rates and contract deals.
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?
Maybe, maybe not.
Have you considered the possibility that the Betazeds and the Deltans are actually the same species and one is just a sub-race/colony of the other? That "Betazoid" might be a more general term like "Humanoid" to refer to otherwise bipedal, quasi-mammalian life forms who possess telepathic abilities?