Were I doing my own Star Trek series, I'd probably do mid-25th century aboard the Enterprise H, though with new content. I envision a twenty-episode season with only one fanwank episode per season, otherwise I'd be doing something completely new or at least new spins on things from before.
I haven't really thought much about the crew aside from the chief engineer being Romulan and sarcastic and rude as all hell to everyone except the Captain, which is why aside from the fact he really is the most skilled engineer around his bad attitude is tolerated. The assistant chief engineer would also be main cast, someone who although the Romulan chief engineer is also rude and disrespectful towards, secretly he does consider him to be a very good engineer. This wouldn't come up until the episode where the crew is exposed to a Naked Time like virus that exposes the true selves. At first the rest of the crew thinks Romulans are immune the chief engineer is still acting the same as he always does, until finally he and the assistant have a scene together and the chief starts being nice to the assistant and praising him, causing the assistant to realize "oh my god, you are infected!" Basically, with the rest of the crew, the chief actually does express his honest opinions of them all the time, the assistant is the only one he puts on an act around.
Honestly, this concept has been in my imagination for close to twenty years now. Only minor changes have been made along the way. At first, I incorporated stuff from the alternate future in AGT about the Klingons conquering Romulus as the chief engineer's backstory, he escaped Klingon conquered Romulus sought refuge in the Federation and eventually joined Starfleet. Since Trek XI I adapted that to having him grow up on a Romulan colony world that got conquered by the Klingons. Also, my original idea was this was on the Enterprise G, because back in the day I foolishly thought there wouldn't be an Enterprise G any earlier than 2450. Since we now live in a time where there is an Enterprise G in 2402, I've decided to go with Enterprise H. This is a ship which has been in service for at least a decade when the show starts.
Obviously it'll never happen, but, this is the hypothetical Star Trek series I imagine myself being the creator of.