I would argue that Berman was the person who got burnt out. If Trek had been handed over to a fresh face who understood the franchise when Voyager ended we might still have good Trek on television today.
Yeah, I don't think Star Trek got franchise fatigue. It just needed new people in charge. Berman should have been gone before Voyager even started, or at least after a season or two when it became clear that voyager wasn't working under him. Between that and the horrible decisions about almost everything in Insurrection and Nemesis, I think it was the people in charge, not the property itself, that put Star Trek in the situation it was in.
Berman and Braga were not the issue. The studios were. B&B have gone on record saying that a lot of things we saw on Enterprise were not their ideas. Things like phase-pistols and the transporter being there from the beginning, was because the studios wanted to make sure there was no mistaking this as Star Trek.
Hell, their original plan was to spend most of season 1 on Earth, getting this ship ready to head out. But the studio-execs were scared it would be to un-Star Trek. So they said, make sure you have all those things people know, so they won't feel alienated.
Same thing goes for a lot of franchises. Like Dr Who. People keep blaming Moffat for everything, but some stuff is just forced into him. "Make sure there are Daleks in this season, or no money to make it." People know Daleks, so the studios want to make sure Moffat puts them in there.
But since Berman and Braga, and people like Moffat, are the faces we know, angry fanboys start blaming them. It's the people with money who ultimatly make the real decisions boys and girls, not the writers.