I loved Moore's response to this very question, if he were to reboot Star Trek the way he did BSG. His response was that there are just some things that are quintessential Star Trek, so a successful reboot would have to incorporate them. Phasers are at the heart of Trek; lasers were not key to BSG, so they could change that into bullets.
While a reboot that goes into more basic and fundamental themes of Trek would be fun to watch, there are just some staples and tropes that can't be eliminated. However, determining those staples and tropes has always been a point of debate among Trekkies. Imo, we need, in no particular order, phasers, warp drive, torpedoes, Horatio Hornblower in Space, space exploration, crew diversity, forehead aliens (though there's always room for imaginative, horrific, or just plain complex aliens), and social commentary in the form of analogy.
I could use less time travel and technobabble-as-conflict-resolution, though.