It was in the movie. In the first movie Spock explains to Scotty that the transporter formula he's using is one that he will invent in the future. He doesn't specify when, but as you pointed out, they weren't using transporters that way during any of the subsequent TV series, so we'd have to assume it was after them.
And then, in the second movie, after Khan uses a transporter to beam to the Klingon homeworld, Scotty complains loudly about how Khan got hold of his transporter beaming equation, letting us as the audience know that this is how he was able to make such a long-range transport.
Okay, so two lines across two movies are used to establish this? And one of them is really speculation on Scotty's part? Not what I call a solid establishment of the truth.
But, fine. I'll allow that the long-range transporter, while genre-breaking, does not stand as great support of how they aren't in the same universe.
It applies to any and all systems of sufficient complexity. It certainly applies to genetics, whose building blocks are specific, but whose outcomes very enormously depending upon initial conditions.
It sounds a lot like you are confusing, "Too many variables to account for," and, "chaos theory." You seem to be confusing "complicated" with "extremely sensitive to initial conditions."
The human body (or any multi-cellular organism) is astoundingly complex. Trillions of cells doing their own thing. However, your system is *NOT* all that sensitive to initial conditions - wake up in the morning after sleeping on your left side, or your right side, and it doesn't really make much difference. The body maintains homeostasis, and continues on largely unaffected by small changes.
This compared to a three-body problem, or a damped & driven oscillator, which are incredibly simple systems, but if you do the experiment twice, with a a couple millimeters different placement, and you get completely different results over time.
Lots of seemingly complicated systems are such that differences in initial conditions are corrected for, or dampened away. Many others are such that a small differences in initial conditions lead to small and/or predictable differences in final state. Neither of these are chaotic, and chaos theory does not apply to them.
None of us have traveled in time and changed the past, right? So, none of us can say if the progress of history is really all that sensitive to conditions.
Nudge those initial conditions even slightly, and a different spermatozoon fertilises the same egg. Genetic result: A person who, if you ran a genetic comparison to their original-universe counterpart, would show as being a twin sibling.
So, is your contention that the overall course of history depends on the detailed content of a person's DNA? That seems a little severe.
We could go very deeply into nature vs nurture here. A person who is at worst Picard's twin is raised with Picard's parents, with Picard's basic life. You think this person is destined to be very different from Picard? I am not convinced.
But, be that as it may, I don't think this will be an issue. I think they are going to (and actually always have) consider the Kelvin timeline *separate* from the Prime, with no impact upon it. I think that was actually the point of having the time travel plot - that they could have a series of movies that *didn't* infringe on their ability to have stories set in the usual canon.
Doubly so now that it looks like the Kelvin timeline movies are done. They aren't going to hitch themselves to a movie timeline that petered out. The movies exist, but have no plot relevance for the Prime Canon.