I disagree they weren't hamstrung by the premise, loss of crew over the years would have had an impact on everyone. Aside from the Pschological impact, what about the physical. Those duty shifts that the crew who had been lost would have to covered, so people would have to do longer etc...
The premise said that they'd spend the entire series going home, that if anyone died they couldn't be replaced and they'd never get any support from anyone ever (no help, no resupplies, no repairs, etc). That means if anyone dies then they can't be replaced at all, not by extended duty shifts or anything.
There was plenty they could have done, I think rather than being hamstrung by the the premise they failed to embrace the premise and all the problems that would come with it.
It's Gilligan syndrome. They had a plot that could never ever be resolved without ending the show and didn't bother giving them any plots that COULD be resolved without ending the show. And the internal conflict was never going to be that interesting either since the Maquis and Fleeters weren't diehard enemies.
They basically took a one episode plot from TOS and TNG (this happened to both Picard and Kirk) and tried to stretch it out into an entire series.