I do find it funny that nowadays it's the grand and unified 'Prime', when around the time of Enterprise it was all 'it's a new timeline spinning off from First Contact'!
It was never officially that; some fans just misinterpreted it that way. The creators' intent was always that
Enterprise would be the history of the Trek universe we knew, not some separate branch. There wouldn't have been much point in doing it as a prequel otherwise.
But, as I said, there have always been fans who see the inevitable inconsistencies and reinterpretations in any new incarnation of Trek as "proof" that it belongs to a separate reality, even though the previous Trek that they see as a unified whole has equally great inconsistencies within itself already. And there are those who mistake differences in real-world production for in-universe differences -- like those who complained that NX-01's technology was "more advanced" than that of Kirk's ship. It wasn't the ship's technology that was more advanced, it was the technology of the TV producers attempting to represent a future starship. The makers of TOS didn't want the
Enterprise to look like it was made with 1960s-vintage technology; that was just the best approximation they could offer of a far-future technology.
Doctor Who is the absolute worst example you could use for trying to argue it's all one neat 'timeline' that leaves the entire fan base happy.
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3) Quiet a lot of the writers flat-out didn't care less about continuity, and simply hoped that the audience wouldn't care if they were having a good time.
Yup. In the original series, first off, they never expected that the episodes would ever be rerun or remembered years later. Reruns were so unprecedented that on the first occasion when they did rerun a serial, they actually had to justify it by setting it up at the end of the previous episode as a recording that the Doctor was showing his new companion. Not to mention that most of the black-and-white serials were erased by the BBC in the '70s, and many of them are lost to this day. There was just no sense of permanence in television back then, and thus no incentive to remain consistent with past stories.
There was also the fact that the show was aimed largely at children. So it was expected that they would eventually "age out" of being
Doctor Who fans and that a new audience would come in, one with no knowledge of the series' history.
On the other hand, changing history and going backwards is Doctor Who's modus operandi. Fuck, remember Genesis of the Daleks? Boom! Previous Dalek encounters are now different to what we saw in the show! And for all its impact on future episodes, it might as well not have happened! Take that fixed point!
I never interpreted "Genesis" as a case of history actually being changed. That was what the Doctor was sent to do, of course, but he ultimately didn't. That was the point -- that he ended up not changing history after all. Any difference in the interpretation of the Daleks was a matter of artistic license and retconning (or Terry Nation just forgetting what he'd done a decade earlier), not an in-universe timeline change.
As a rule, classic
Doctor Who wasn't concerned with time paradoxes or changing history or any of the preoccupations of most time-travel fiction. Usually, it just used time travel as a means to deliver the characters to a new setting, no differently than how
Star Trek used space travel. Only a few stories in the original series really did anything with the questions of changing history or the creation of paradoxes or weird temporal phenomena. It's only in the modern era, thanks mainly to Steven Moffat, that time travel itself and the paradoxes and causal loops it creates have become a central focus of the show.