Also, I have to wonder if the timeline really matters at all.
... how much would really change if it was set in the 23rd or 24th or 25th centuries?
... Other than a few throw-away lines here and there, there probably would not be much if any difference between a Prime show and one set in the NuTrek reboot.
The state of Vulcan would be a huge difference.
And those throw-away lines
could have meaning to those who understand its reference. I'm only using The Avengers as an example because I literally just saw it but there's a scene where Nick is telling Thor about the scepter that's able to turn people into the users own personal flying monkeys. Thor didn't get it but CA ringed in that he got that reference, and he felt great about it. The rest of the movie continues and those who got that scene, got it, and those who didn't, weren't going to storm away from the movie because they didn't. Plus a 25th century series would be foolish not to reference to prior series. When Admiral Jones tells Commodore Teton "We expect more casualties then the time Sisko retook DS9," those who got it would smile, those who don't could imagine it was a big battle, and those who want to find out can buy the (soon to be released HD) episode and see it for themselves. It'll be a source of revenue.
IMO throw-away lines gives stories replay value.