Christopher, are you referring to Return to the Planet of the Apes? That was not an inferior adaptation, even though it didn't last long. The others you mention and many more were, however, but Apes and Star Trek seem to be rare exceptions that continued from the other versions.
Well, such opinions are subjective, but I don't think the show was that highly regarded. It had some interesting ideas and a surprising amount of continuity, and it was progressive in featuring an African-American co-lead, but the animation and acting were pretty weak, and I don't think it had a good reputation at the time, even if nostalgia has been kind to it.
And it didn't continue from any prior version. Like the just-ended live-action series, it was an alternate take that drew on elements of previous versions but wasn't consistent with them. The movies and the live-action series both featured a pre-industrial ape culture, but the animated series' apes had technology equal to 20th-century Earth's aside from being backward in aviation.
As for other cartoon spinoffs of live-action shows, most of them
were purportedly continuations, even if they changed some details or went off on bizarre tangents that the live-action shows ignored. For instance,
The Fonz and the Happy Days Gang were meant to be the same characters from the prime-time show with the same personal histories; they just happened to meet "a future chick named Cupcake" and get swept away in her time machine. And
The New Adventures of Gilligan picked up pretty much where the live-action show left off, with the only major format change being that Gilligan's baby-chimp sidekick was a regular character (though such a chimp did appear in at least one live-action episode) and that Gilligan was more often the lone innocent who was immune to the others' flaws and vices, rather than the bumbler who screwed up their rescues (but there were live-action episodes like that as well). And the infamous '80s revival
Gilligan's Planet was itself a continuation and sequel to the previous animated series.
As for the OP, well, doom and gloom aside, it might not have been as terrible as it seems but it's hard to see how. It is good that isn't the way they went.
Well, the intention was different with this idea. It was meant to be a show for young children that would be a counterpart to the live-action show for adults. It was conceived at a time when TOS was still on the air, so it wouldn't have made sense to try to duplicate TOS exactly; instead, the idea was to offer something complementary, geared for a younger audience. But TAS came along after TOS was gone, and so it was designed to be a direct continuation, the first (?) Saturday morning cartoon specifically made for an adult audience (although still suitable for children).