What Trek trappings could be removed, but still allow the show to be Trek?
That's actually an interesting question.
Given TNG as the prime example you could theoretically create an entire new cast and perhaps even have them aboard a new ship. But I think you do need to have an interstellar traveling ship with FTL capability. Hence the "star trek."
It's been mentioned before but ENT missed a huge opportunity to give Trek a fresh face. What they ended up doing was half-hearted.
So it comes down to how creative are you willing to be? And how willing are you going against popular expectations? This forum has been full of ideas on revamping Trek in vatying measure from mild to wild. The only guarantee is that some will love it and some will hate it.
I think a big part of it would be the kind of stories told and how they're told. I also think any new Trek has to have a measure of optimism to it because that is something that is really identified with
Star Trek. Making it
too dark and edgy and outright cynical would be something too drastic for a lot of people to accept. If one were to go that route then you might as well do something non Trek where there would be no expectations.
Although I never watched
Andromeda beyond a few early episodes the concept was essentially akin to showing a heroic group aboard an advanced starship trying to restart the Federation. In
some respects
Babylon 5 felt more like the kind of show ENT could have been in showing us a pre TOS era.
I don't think it's so much a question of taking away familiar trappings as opposed to reinterpreting them. JJtrek is, for all intents and purposes, a very mild reboot conceptually. It's basically a tweaking of the familiar and dressed in still familiar trappings. It's most drastic change was in the how it told its story.
But one could reboot/reinterpret
Star Trek in a more drastic fashion. You could still keep many familiar elements yet reinterpret them in a much more contemporary manner. Although it may seem dated now (in some respects) this is what TNG did essentially. The key distinction could be to cut away from the established continuity. Sometime ago I suggested such an approach for a
Star Trek that reinterpreted TOS and TNG into an entegrated whole. There was still an
Enterprise, a Kirk, a Spock, a Picard, a Federation, Klingons, Romulans and such, but it was all reinterpreted and merged into a cohesive whole. The Next Generation literally was the
next generation that gradually takes over from Kirk and company aboard the same
Enterprise. Many of the familiar characters were there, but reinterpreted (and some with gender changes). By the time our new Picard has command of the
Enterprise our new Kirk is an Admiral. If done just right it's an idea that could run for ten years as you transition from one group into the next.
For what you could take away you really only need to look around at what else has been done in the visual medium as well as in print. Lots of people have taken the basic idea of
Star Trek and revamped into something different to suite their own tastes. Admittedly they make it work partly because it's no longer called
Star Trek and so it's easier to allow for broader changes. David Gerrold's
Star Wolf books are essentially that.