The STARGATE TV franchise gave us 10 years of SG-1, plus additional series in ATLANTIS and SGU. That was a 14-year-run in a single continuity. Granted, STARGATE on TV wasn't exactly the same as STAR TREK, but continuity obviously wasn't the problem there; the newer series simply weren't as good as SG-1, so the stories didn't measure up, and the ratings declined.
LAW & ORDER ran for 20 years on NBC, spawning multiple spin-offs. While continuity wasn't perfect, it was mostly contained in the same "universe" with SVU still in production.
Why should continuity be seen as a negative? It's what Hollywood does with a given show, its creative agenda in terms of what stories it wants to tell, that makes a show appealing to an audience. When TOS was on NBC and later in syndication, it was obviously known for its commentary. It seemed that, with each new series, and especially the reboot, STAR TREK has become less oriented toward commentary and more toward action-adventure/explosions and further away from commentary. It seems, more and more, that Hollywood wants to milk TREK as a cash cow but also shy away from its roots.
There's nothing wrong, in theory, to remaking/rebooting TOS but if you're unhappy with the kind of storytelling that TOS started to do then why remake/reboot TREK at all?
For decades, soap operas have thrived on American daytime television, frequently producing years and years of episodes, five days a week, in a single show's continuity. It works. It's profitable. I doubt you would often hear anyone saying, "Let's cancel this soap, wait a few years, and reboot it with all-new cast and unrelated stories under the same name. The audience will think its fresh!" Seems unlikely. It seems more likely that a network/studio would cancel an old soap and replace it with an entirely new one that has no ties to the original.
If you think the daytime soap opera analogy is inapplicable to a dramatic TV franchise like TREK, guess again: DALLAS and KNOT'S LANDING were on CBS spanning multiple decades, DALLAS starting in the late 1970s, both on through the 80's and clear into the 1990s. They commanded a large audience, even through creative stumbles, and never jettisoned their continuity.
That's not to say that continuity is always precious and never to be challenged in any way. Actually, VOYAGER had the opportunity to renew the STAR TREK universe and the VOY makers obviously screwed it up. Starship Voyager was supposed to stumble into a trap that teleported both the Voyager and its Maquis adversaries across the Galaxy, where no familiar faces or past storylines ever to be seen, right? Well, then we see Voyager doing battle with the ill-conceived Kazon (who cares?) and when that didn't work, it's time to bring on the Borg! And we even wound up seeing the Ferengi from TNG's "The Price". (You remember the Ferengi from "The Price", don't you? Nah, neither did I.) So, what good did it do to show Voyager being snatched clear across the Galaxy in the first place? If VOY was supposed to represent an attempt to refresh the TREK universe by plucking a starship out of familiar space and showing it in an unknown void, they should've sent the ship to a completely different galaxy with no hope of ever returning home. Then you'd have to start over with a clean sheet of paper.
And what did the JJverse give us? A reboot of TOS, with a tattooed Romulan nemesis. This represents a "new" continuity? Looks like a new and improved VOY instead.
If continuity is such a bad word, wouldn't it be better to reboot STAR TREK with a completely clean sheet of paper? If Hollywood can't do that, then there really isn't anything wrong with the Prime Universe in the first place. It all depends on what a creative team wants to do with that Prime Universe. So far, none of the spin-off series or movies exhibited any coherent creative agenda that took the franchise, comprehensively, in a specific direction. Direction is what is needed, not repudiation.