This used to bug me as far back as the 90s - that Trek shows seemed to need three seasons to get right. Most successful shows may or may not get better with time, but rarely get the luxury of waiting til their third year to get it right.
TNG had the '87 writers' strike, and VOY and ENT were both rushed out of the gate prematurely. But it didn't help the cause.
Yeah, TNG was probably lucky it was on syndication. That maybe gave it a bit of breathing room. And it was something brand new so I think Paramount seemed more open to giving them a chance. I noticed some improvement in season 2 in fact and by season 3 it seemed to really catch fire. Perhaps that's why DS9 was given time to find its feet. TNG was still hugely popular and it paid off with DS9. Voyager was probably the most consistent of the shows in that I didn't notice that first 2 season lag there. But I think that's where the Berman fatigue really crept in.
That combined with the relative unpopularity of the films Insurrection and Nemesis I think also put Enterprise on a short leash. Enterprise wasn't given the same honeymoon as the other shows, and it was a network show so I think that shortened the leash even more. I do give Berman credit for shaking things up in seasons 3 and 4. He was not a great writer or even creative mind, but I think his strengths were definitely more on the production end of things. Bringing in the right producer to shake things up (Manny Coto), changing the focus and doing multi-episode arcs, something that was just beginning at that point on TV. And he finally did something he should have done a few years prior, take on more of a back seat role. He was fine as an overall CEO. Making sure people were doing their jobs, overseeing budgets, and just making sure the show overall stayed on track. But he should have left the creative decisions behind a few years prior, maybe during Voyager's run.
But in all honestly Enterprise's fate was probably already signed by Moonves. From what I've heard he probably needed little reason to cancel a Star Trek show. Enterprise's issues in the first 2 seasons may have helped make the decision, but I think Moonves would have just used a different excuse to cancel it.