You do need aliens and exploration to be Star Trek, and you also need optimism. But, I think you could scale back the tech and still have it be Star Trek.
Warp drive is a plot necessity in order to make it possible to visit other planets. Transporters could be removed, or you could make it so for transporters to work you have to have a transporter fixture at both ends. You can remove all other god-tech to give it that BSG machine gun aesthetic. You can also make the aliens less humanoid and less like us, like the ones in the Ender series. Just so long as the basic drive of the show is to go out, explore space, and try to make friends with aliens.