While you may have an argument there with Enterprise, that certainly does not in any way explain why Voyager was such a weak, anemic series. Nor does it explain why DS9 was such a great series. (Or did the tipping point of when continuity became an "unbearable burden" occur sometime shortly after DS9 ended?)
As I edited in later, a series set further in the future then VOY would also not have worked.
We already have transporters, quantum slipstream and what else; it's the same problem that the Stargate franchise now struggles with. If the technology is so advanced as to almost be magic, what would be left? No danger, that's for sure. You can't simply make the antagonist bigger and badder each time, you know; you'll end up with godlike beings who eventually would be dumbed down in order to defeat them.
And without danger to the characters, no urge to evolve. That would make a boring show. A new TNG set in the far future would also not have worked. Either we've seen or heard everything already (most episodes of every series was a rehash of an earlier one), or things would be so unfathomable as to alienate any 'regular' viewer the series would have left.
The continuity became an unbearable burden at the point where shows in the future wouldn't have worked, shows in the past didn't work and the time in between was already claimed.
Really, though, this just ties in what what I feel is the fundamental fact you are missing; that Star Trek is entirely capable of growing and expanding and reflecting changing realities without having to go constantly back to the beginning. Why should we go from Kirk to Picard to Sisko to Janeway and then back to Kirk? Why not a captain beyond Janeway? And then one beyond that one? To borrow from your gardening examples earlier, why not let the annuals that are Star Trek grow again and again, rather than tearing them out and planting a whole new set each season?
As I said before, because of the continuity. There are lots of purists who seem to regard that continuity as holy. Therefore, you'll have to advance the technology as well; even up to a point where it all comes unraveled.
Stargate has the same problems. Technology better then most enemies, so the galaxy spanning Goa'uld empire is defeated. Then, you need a bigger baddy: enter the godlike Ori. And better technology (Ancient and Asgard stuff) and better allies (ascended beings) to defeat them. There's not much beyond godlike beings you can get.
Even if you do not focus on the technology for a moment, you'll have to admit that most episodes of most series have been rehashes of earlier episodes. Only, with the added continuity, people could not 'get into' the series anymore because it was assumed they 'knew stuff'. Which they didn't. As such, the number of viewers dwindled until it was unprofitable to continue, essentially declaring Star Trek dead.
Anyway, that's beside the point. It's not that hard to keep continuity errors in fiction to the same incredibly minor level as someone in the real world claiming that Columbus was the first to discover the New World, or that Canada beat America in the War of 1812. Heck, five minutes to check a date or a reference on Memory Alpha would do it. Nerds are more than happy to collate all the little details from works that they love, and thanks to the internet there's no reason whatsoever that the creators can't have access to a relatively professional history text for a fictional universe.
Yes, but the continuity errors had already exploded into a major level. And the crosschecking isn't that simple. For a single fact, perhaps. But all facts needed in an episode? Most writers aren't Trek addicts; they are screenwriters, they do that for a job. Not only for Star Trek, but for a boatload of series. They can't afford to waste all their time looking up silly little facts, they need that time to be creative.
And as far as VOY goes, it does in fact mean that the writers were lazy. Voyager was on the far side of the galaxy, dealing for the most part with things that had absolutely never been so much as hinted at before. And what did they come up with? The kind of 'encounter of the week' stories that were getting bland and predictable when TNG was wrapping up it's run. VOY could have been an awesome series, full of action and adventure, romance and comedy. Heck, it could even have had the titular ship come home once DS9 was done and have adventures in the Alpha Quadrant where things evolved and changed, in radical ways. But it wasn't, and it didn't, and the fault lies entirely at the feet of tired, lazy writers who simply didn't have it in them to do anything really new and challenging, and a licensing company that didn't actually want them to do anything but the same old, same old.
VOY wasn't, because people were used to a certain continuity; also in the level of writing. You don't simply blow up a planet. You don't simply let Voyager be destroyed, letting the survivors evolve. The hardcore viewers don't expect that sort of thing. If they did, it would be wonderful. And if they did, fans would be breathing down their neck claiming for revenge faster then a speeding bullet. "Year of Hell" was about as far as they could go; provided they had the reset button handy. And it was a shame, I agree with you on that.
Star Trek had a certain image, and in order to retain the small amount of following it had left, it could not change that. The fact that this reboot has changed all that is all the more astonishing.
That's why stories should be based on characters not technobabble. Or transwarp beaming.
Here you hit the nail on the head. Stories should not be based on technobabble.
But because the technology had become so integrated into the Star Trek universe, people
expected solutions like "just inverse the polaron-magneto-coupling to extrude the planarized-subspace-rift". Or even something simple like "it seems they're in danger. Beaming them out now". Because if they didn't, people would complain why they didn't simply beam the characters out of the nasty situation they were in. So the writers tried to explain why beaming didn't work with lots of technobabble to satisfy those viewers. And then there's the technobabble again. It's simply what Star Trek had become.
Star Trek had become a "Damned if you do, damned if you don't" situation for the writers. For people had already seen someone beam out of a cave. Logic suggest they can do so again. So they had to make up lots of technobabble reasons in order for stories to work. And that was a shame.
Not about it's boring bunch of knock offs, which drowned their stories and characters in Treknobabble, and eventually bled the franchise to its near death. TNG lasted 2 seasons beyond what it should have, and gone out with quality not quantity. 
