I think that if anyone ever wants to do Star Trek- most realistically for an RPG campaign- they'll have to sit down and decide what elements of it are canon and which elements are not. This doesn't even have to be due to politics (though it will be if dealing with anything from the last 10-15 years) but simply the inevitable drift and sprawl a decades-old franchise that's been through the hands of multiple writers who didn't consult or research with one another to keep things straight. Everything made by Star Trek from the end of the Kirk era has been mostly written by people who don't know the minutiae of the lore like a hardcore Trekkie does and and canon that's been added to enough begins to contradict itself.
Even within the Kirk era, there are tons of internal inconsistencies. The original and animated series had all the big problem issues: warp drive time travel, open-ended time travel, transporter duplication, saving and changing in mid-transport, and so forth. They got some great writers like Larry Niven and Harlan Ellison, but the writers often didn't know the lore like in later series.
When I ran a Kirk-era Star Trek campaign, my approach was that *all* the episodes were roughly as true to in-game reality as believable "based on a true story" dramas. So no details of episodes were guaranteed real, but the broad points were correct. I avoided time travel entirely, like the OP suggested. There were many rumors of time travel but it was similar to Area 51 and aliens for the real world. It was theoretically possible, but any use of it was either a hoax or very well covered up.
One of my main points to re-work in any Trek canon I'd control would be hand phasers and the nerfing thereof. Back in TOS, a hand phaser was a horribly deadly weapon. Phaser combat usually consisted on one person pulling a trigger and the fight being over- buildings could be blown up and crowds rendered unconscious by a single shot from a phaser.
It's one of my pet peeves, and players in the ST game I ran were sometimes shocked to find what hand phasers were capable of and how short and deadly phaser combat was. It's a good thing the Modiphius game has metacurrency because the way I ran things it was "whoever doesn't shoot first, spend metacurrency to not be removed from the fight".
Yeah. Once you're hit with a phaser, you're down was my rule for Star Trek. There were still a handful of exceptions in the Kirk era, like the Horta, but they are very rare. I would sometimes feature extended hand-to-hand combat, but phaser combat was very quick.