I ran a Star Trek RP set in 2080.Ultimately I think we mean different things with 'continuity.' To me continuity cannot have contradictions, canon can.
My thinking might be affected by running a lot of RPGs, I tend to think setting as a coherent, consistent whole. So If I'm running a Star Trek RPG set in 2262, then the question "does Starfleet have knowledge of cloaking devices" can have only one answer in that setting. As setting where there is a different answer to that question is a slightly different setting with a slightly different history.

But when it comes to film and television, you have to accept some inconsistencies, and that gets harder to control as the franchise ages. Reboots are one answer. Hollywood loves them. It frees up writers to do what they want. They can use what worked in the past or try new elements.
Star Trek fans are.. uh.. not always the most pleasant souls. 2009 was a reboot of sorts and there's just no pleasing some fans. Should they do it again? Honestly I don't care, as long as the writing is good and they have good actors and the spaceships look like star trek, which I'm not really that picky about either. But if they do, it will won't please everyone. They should just keep doing what they're doing. It's working.