...and running multiple series at once on TV fatigued and nearly killed any new Star Trek.
I still say that "franchise fatigue" is a false argument. Outside of DS9 getting some middle-child effect from lack of corporate support and interest, I think the viewership problem for Star Trek was really just a quality issue.
<rant to follow>
I know there are people who started with Voyager or Enterprise who say that those are their favorites or the ones they think are best, but I think the vast majority of fans, critics, and general viewers would agree that VOY and ENT are not the franchise's best shows. Issues include:
1) Voyager tried to replicate TNG's adventure of the week for syndication and out-of-order rewatchability mostly - avoiding having to consider the consequences of past events which DS9 included regularly. In this way Trek wasn't keeping up with developments in television overall, and I think it frustrated regular viewers. Trek didn't really fix this until ENT season 3 (which was mostly serialized plot continuity) and even then could have used more character growth and continuity.
2) Both Voyager and Enterprise failed to really follow through on their premises due to corporate interference and some producer/writer laziness. Voyager dropped the inherent conflict with the Maquis crewmembers starting with episode 2 - except for two episodes: Learning Curve and Worst Case Scenario. Little effort was made to really engage with the "lost and desperate far from home" elements except for lip service paid to having to occasionally find more food supplies. Issues with unending power supplies, torpedoes, shuttle craft, and always running into the same crappy bad guys (the Kazon, for two whole seasons despite going "full speed" for home) demonstrate lack of follow-through. Enterprise hardly seemed to be showing the true early pioneering days of space travel since they picked up phasers, transporters, hull plating (i.e. shields) and all the other tropes in short order. They also had corporate interference with the shoehorned Temporal Cold War which went nowhere, immediately putting the ship into space in the pilot instead of exploring the organization and forming of the first starship and crew, and that "finale".
3) Voyager also had clear and apparent issues with lack of care or interest in some of their characters by both writers and actors. We never learned much about or saw much evolution for Chakotay and Kim, the writer's apparently had no real future plans for B'Lana as late as "Barge of the Dead" (according to Ron Moore). Enterprise did somewhat better with trying to develop all their main characters (at least at the start and given they only had 4 seasons), but I don't think many people count Reed, Mayweather, or Hoshi among their favorites, or would list them as well-developed or even noteworthy. If the character wasn't currently compelling, the writers should have thrown a curve ball or something at least trying to engage with the character!
TLDR
Ultimately I would say that being too conservative was the problem for the fall of Star Trek. And despite all the good that Berman did to hold the franchise together and keep true to its spirit, the restrictions he and others placed on it, and the lack of new blood at the top after 17 years of continuous production really hurt. Not investing in the shows and their characters, not taking enough risks, and not evolving the show with the times - as weird as that seems for a forward looking franchise like Star Trek - was what led to the downfall.