Star Trek's core becomes clearer when you contrast it with dystopian science fiction, ranging from
Nineteen-Eighty-Four to Cold War paranoia movies like
Invasion of the Body Snatchers, which depict the future and outer space as frightening, uncontrollable and negative, or
The Day the Earth Stood Still, which questioned whether humanity was worthy of survival.
There really wasn't much in sci fi TV or movies that was both positive and intelligent. The positive stuff previous to
Star Trek tended to be Flash Gordon/Buck Rogers fluff.
Star Trek's innovation was to merge a positive message with an underlying intelligence that gave the positive message some credibility.
Star Trek tells us that the future and outer space can become a warm, fuzzy extension of liberal American ideals of tolerance, democracy and freedom. Even mean aliens like Klingons can be brought to heel by the all-encompassing blanket of the Federation. Of course, once in a while, shooting wars will break out. These are only interludes, puncutating the inexorable advance of Federation ideals.
All the series didn't advance this idea to the same degree, but to the extent they had ideas at all, they did.
TOS - Strongly advanced these ideals. Kirk & co's whole job description was benign imperialism, to defend Federation territory and expand its influence wherever possible.
TNG - Advanced these ideals in a less overtly imperialistic context. But Picard's nannyish lecturing of aliens that didn't hew to Federation ideals got the point across regardless.
DS9 - Implicitly advanced these ideals by depicting them under severe strain of serious warfare and bending but not breaking.
VOY - Janeway continued Picard's nannyish behavior, but in general, VOY shied away from opportunities to advance Trek themes, such as demonstrating how they might be applied in a real conflict between the Starfleeters and the Maquis, not because VOY was opposed to these ideals but simply because it didn't seem much interested in ideas other than the most surface ones.
ENT - Archer managed to be nannyish even tho the Federation didn't exist. I guess he was just ahead of his time. The fourth season delved somewhat into how the Federation, and its ideals, might have come about.
So reboot
Star Trek all you want. If you tell stories that advance the core theme, it will be
Star Trek still. You don't need to use the same characters or even have them be in Starfleet, but then I have to wonder what you're rebooting?
Pretend that you've been given the task of spearheading a total reboot of Star Trek, bound only by the franchise's underlying theme(s). What arcs do you use to sustain the narrative, and how does said underlying theme factor in?
Sounds like you're not doing a reboot, you're just extending the theme to other situations. The mind boggles at the possibilities. You have many centuries and a whole galaxy to play with! To me a reboot means you're discarding existing canon, but you could come up with stories that don't touch on existing canon other than just the broadest stuff (Starfleet exists, who the various aliens are, etc.)
Just off the top of my head:
-Series revolving around Federation diplomatic core.
-Go back to the Medusans from TOS. They are now Federation members, but they still are fatal to humanoids. How do they integrate into a humanoid-centric organization? Can they join Starfleet by wearing some sort of humanoid exoskeleton and will they be accepted by other species? A test of Fed ideals.
-Continue the Spock/Rommie unification plotline. How would unification between Vulcan and Romulus impact the Federation.
-ENT never really did the Birth of the Federation well. Go back and tell that story.
What I'm talking about is a total re-imagining of the franchise starting from the ground up, retaining only the theme of 'exploring the human condition'.
Exploring the human condition is way too broad to make it
Star Trek. All literature explores the human condition. You might end up with a nice space opera story, but it wouldn't necessarily be
Star Trek.