But none of them work under the illusion that Supergirl (which I like), is somehow in the same continuity as the Superman films or the 1950's Superman TV show, or any of the comics. It is its own version of the universe, not a continuation.
Well, of course DC has its' "multiverse" explanation for differing continuities, so of course fans have that to fall back on.
But even with that, as I stated with the original Superman example, frequently die-hard fanbois of one sort or another will cry about a new version, crying that it isn't the "real" Superman and so on.
But the overwhelming amount of complaints that I've seen on these forums about Discovery haven't been about continuity, they've been because the show doesn't match up with the tone of what the complainer considers "real" Star Trek.
But kind of like with DC's multiverse, if a fan really hates Discovery and likes the version of some other show, they really shouldn't freak out because the likelihood is that eventually a new version they like will show up.
Back with a Supes example, many fans were very upset about Man of Steel's darker tone, including Superman killing a villain. It's a valid complaint, but guess what? Boy Scout Superman showed up again, the people who make these shows will produce what sells.
In this clip, you can see they even made the character use outdated catchphrases as a shout-out to fans that were disappointed by the last movie version of the character.