I dislike time travel when it's used too much, it opens a huge can of worms. The 29th century federation time shops for example, who takes care of what for example? If they scan time and notice Voyager mucking around on 20th century earth who, does a time ship from march 2843 or one from july 2844 go? Now maybe those events are already part of history and they've known which timeship goes since the program was started in 2812 but in that case it would be ine of the most boring jobs in the universe. They'd just play out "scripts" the time continuum has written for them. Even careers would be affected by it, "sorry Braxton, we can't promote you, if six months you're supposed to travel back in time and become a hobo for a few decades but don't worry, it's an anorted timeline, you won't remember it, you just have to do it. Thx bye"
So yeah, an Enterprise could be a timeship but I wouldn't want to watch that show, not with Star Trek's tendency to explain things because the more you explain the logistics of time travel the less sense it makes.