I voted Kosinski - he's my logical choice. Brought back Tron to a modest success (and a decent movie if not spectacular) and Oblivion was better than it had any right to be. Both were visually gorgeous, and Oblivion in particular looked like JJTrek, just more restrained. Plus he hasn't committed to any project as of late, but is supposedly circling other 70s sci fi reboots like Logans Run.
That being said - I would like Joe Cornish to get the gig, if only because of his work on the Adam and Joe show taking the piss out of TNG (lovingly) back in the mid 90s. This guy is a fan who loves it but knows the camp factor well. If he wrote a script, personally I'd feel it could have the best balance of Trek-ness to action movie you could get.
Edgar Wright could work. The constraints on Ant-Man of tying it into the larger MCU shouldn't have any relevance here, as for all intents and purposes 3 is the last one to round out the trilogy. I can't see the actors hanging around after their contracts are up, especially Zoe Saldana. That being said, EVERYTHING Wright has done has been a passion project. Everything has taken forever for him to write and get to production. Writing a script in a proverbial weekend and shooting it the next day just doesn't seem like his style, and I don't think Trek was ever such a big deal for him. That in and of itself isn't important, because lord knows JJ wasn't a fan either but JJ had years of lead in and complete control. Wright would be jumping onto a steam train and shovelling coal all the time. That just doesn't seem like him at all.
Frakes I'd like to see, just like I'd like to see Bay do it. It'd be an interesting experiment either way.
And while i'm at it, my total pie in the sky wishful thinking shot would be Aaron Sorkin, or Sorkin/Fincher collaboration. Sorkin has an obvious passion for Trek, as his shows always seem to remind me of it and his "last ever TV show" (or so he says) is over with next week with nothing to my knowledge announced for him that hasn't already been in development hell for a few years.
So TL;DR - I like Kosinski as the safe bet. I like Cornish too. Wright would be good but I dont think would work, and Sorkin/Fincher would be amazeballs.