If there's a Trek series in the present day, I'd rather it be in the present day in the Star Trek Universe--the one in which it's only been twenty years since the end of the Eugenics Wars and the mysterious disappearance of Khan. A present day Earth where Sanctuary Districts are becoming more common around the world, much to the dismay of many. The shadows of World War III are still a few decades away, but new international factions are being moved in that direction. In space, however, there are sleeper ships on long multi-year voyages to Alpha Centauri, but likely a number of established settlements and space stations throughout the home Sol System that are only days apart at impulse.
Rather than trek across stars, the show would trek across the various settlements in the Sol System, with the troubles on Earth being something that raises its head periodically for our heroes.
Screw the real world.