I could live with more than one trek show at a time, but they'd need to be good, and diverse. DS9 and Voyager got away with some mediocre episodes, but there wasn't a lot of competition, not of the on demand magnitude that we have now.
I think three thematically different shows could coexist if they were done well and didn't rely on the old clichés.
A battleship series, engaging the enemy. Patrolling the frontier. Much like discovery.
An explorer ship series, investigating some great mystery of the universe, something akin to the Chase, but told over three or so years, with many twists and turns.
And then something less conventional, set on a star base, or even multiple locations across the galaxy, an epic conspiracy story maybe, again, told across several seasons.
Three vaguely inter connected series, produced concurrently, but only one releasing episodes at any given time. That could work.