It’s not me who took the approach, it was the (many) writers of most of serialised television for the past sixty years who did. And until very recently, the audiences didn’t give much of a shit either.
Is it that they didn't give a shit, or that they didn't have the means to?
More exactly, it's a given that TV today is more serialized than it was when stuff like Doctor Who and Star Trek first came out. But part of the shift is that with stuff like the Internet, streaming, etc., it's far easier to stay up to date with continuous storylines. So with that in mind, I don't know that it's the case that people from decades ago didn't care about continuity, since they'd probably understand that it was harder to maintain continuity.
But even then, there's a separate issue to consider. Stand-alone episodes is fine, discrepencies between those episodes are still discrepencies. Using TOS as an example, if in one episode Spock says "I'm from Vulcan," and in another says "I'm from Heaphestus," then that's still a discrepency that anyone with reasonable interest in the series is going to pick up. The fault would still lie with the writers.
Rowling is kind of notorious for continuously tweaking the lore of Harry Potter outside of what's been established in the novels.
Yes, Rowling's recontextualized/added stuff, sometimes clumsily (e.g. Dumbledore), those aren't discrepencies ipso facto.
Off the top of my head, I know McGonagall's in a weird place right now thanks to 'The Crimes of Grindlewald', but not much else. Yeah, Cursed Child has some weird shit in it, but it's not outright a breaking of the setting.
"Fair" is something of a subjective value.
For a show like Doctor Who with dozens of writers across hundreds of episodes and decades of show lifespan, I would argue actually no it isn't fair to expect an encyclopaedic knowledge of everything that has gone before from writers. I think you could however consider degrees of inconsistency: few if any in TV series are big ones, they're more the lore equivalent of someone accidentally wearing a wristwatch in the shot of a medieval epic. (If that's the sort of thing bothering some people, these people need to go out and get a life.) Never mind that most writers think they don't think consistency should get in the way of a good story.
To reiterate, I wouldn't expect a writer to know everything off by heart within their head. I WOULD, however, expect them to remain consistent with what's come before, whether it be through consulting the Doctor Who wiki, past episodes, whatever internal story bible the BBC might maintain, etc. Even if Doctor Who is a setting where discrepencies can be explained due to how elastic the timeline is, some discrepencies are more...discrepenful than others. To use your example, a watch on someone's wrist isn't a deal breaker. On the flipside, take Flux, which among other things, destroys (or at least devastates) the entire universe bar Earth in the 2020s, so how does that affect everything that occurs in-universe after that point? Don't know, I doubt Chibnall knows, I doubt he particuarly cares either, and in case you're wondering, I think Chibnall's run has been mostly terrible, and that's part of the reason why.
As to the question of what's "fair," or the notion of writers not letting continuity get in the way of story...okay, I'm going to use a personal example here. I've tried to avoid examples like this, but this thread's already got quite personal, so I can go the extra mile. So here it is.
The last multi-chaptered story I wrote on FFN was a Starship Troopers story that by my count, is around 70,000 words. Since the Starship Troopers wiki is pretty sparse and poorly sourced, it wasn't something I could rely on. So, to ensure that I kept to canon as much as possible, among everything else, I did the following:
-(Re)watched the five movies.
-Read the movie comic adaptation, plus Insect Touch, plus Dominant Species.
-Read the script and took notes of Terran Ascendendancy, plus scans of its game manual.
-The same for the manual of the 2005 Starship Troopers game.
-Skim-read the original Heinlein novel
-Kept a codex of sorts for the story, cobbling the stuff together into something as cohesive as possible - key dates, technology, characters, chain of command, etc.
You might point out, among other things, that I could have done more. And that's true. You might also point out that those pieces of media aren't always in the same continuity and that's also true (for instance, skim-reading the novel was mainly about quote mining, so the same quotes could be given as a form of subversion, since the story's set in the films continuity). But the point is that if I can do all of that on my own time, for free (similar to wiki editing), then it's not unreasonable to expect people who write/produce for a living to put in something resembling the same effort, and frankly, more.
For instance, before directing Wrath of Khan, Meyers watched every prior TOS episode, whereas Baird refused to watch any of TNG before directing Nemesis. The quality/lack of both films probably wasn't contingent on that, but ask anyone which film turned out better, and you'll get only one answer. At the very least, Meyers was putting effort in, whereas Baird wasn't. Nothing in Khan really clashes with TOS, while people have picked apart the insanities of Nemesis from a lore standpoint. And that's still before you get to the story aspects.
Congratulations on being exceptional in this way.
That's hardly exceptional. If anything, in my experience, NOT caring about continuity is the exception.