That's an interesting idea, but also brings up questions of it's own. Kirk explicitly refers to himself as a soldier, indicating military background. The Intrepid is a Connie - although that could be a case like the Arleigh Burke / Kongo classes today - and how do we know when the E is on UESPA authority vs. being forwarded to the UFP.
Hmmm, answering all that and tinkering with the tech and background would let you spin a whole new SF universe out of this...
Yes, Kirk, at the time he says he is a soldier, was right at the start of the Federation-Klingon War and the Organians identified him and the Enterprise as "military forces". I personally think that the Enterprise operates under UESPA which provides forces to the Federation. Vulcans, Tellarites, Andorians, etc probably have their own fleet authority that provides ships the Federation in TOS.
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.
Multiple-universe/alternate continuities could easily have the same characters with imperceptible to radical differences. For example, the Mirror Universe had the same characters and even a near identical ship but the galaxy and the characters' upbringing and motivations are radically different. And then there is also the Antimatter Universe where people and tech are practically identical except for the reverse charge. We also have "Parallels" from TNG where they had thousands of alternate universes pop up their own Enterprise-D but each ship had different histories, like one where Picard died at Wolf-359 but they still won, or the one where the Borg had overrun everyone and they didn't want to go back. You can also look at the Stargate SG1 series as they handle multiple universe/continuities to see it isn't that difficult from a story handling POV.
As to where to break the continuities it depends on how detailed or broad you want to be in identifying a focal point. I'll give you some easy, broad strokes to think about
James Kirk's fate. In TNG's "Relics" Scotty says he was alive when he retired. In "Generations", Scotty sees Kirk killed-in-action. Right there you already have a splitting point.
Abrams Trek and Enterprise's "In A Mirror Darkly" claims to be from TOS, but as you can see, which TOS? The one where Kirk dies in "Generations" or the one where he lives in "Relics"? Or was it from another alternate TOS universe called the "Prime TOS-verse" which might not be the same as the original TOS?
Earth's last World War. In TOS, it was identified as the 1990's Eugenics Wars with casualties of 30 million. In "First Contact", World War 3 occurs in 2050s with 600 million. In "Doctor Bashir, I Presume", DS9's Eugenics Wars takes place in the 2100s.
Voyager's "Flashback" shows Tuvok as part of the Excelsior crew and Valtane dying whereas "The Undiscovered Country" shows Valtane alive at the end.
As you can see, it's alot easier to put each series (and even movies) into their own universes and consider them all as alternate universes. The original TOS could have had the TOS Movies follow but things could be quite different in their far future. They might even have had a TNG but details and technology would've been different. Conversely, TNG's pre-history would've had a TOS but the tech and details would be slightly different. TNG Kirk could've heard of Iconia but TOS Kirk might not have.
And then there is the universe from "First Contact" which spawned "Enterprise" which led to AbramsTrek. Etc, etc.