Bruce Arena is basically the best manager in MLS history, no (5 MLS cups)? However, lumping Nicol into the others is a bit unfair to him I think and is basically based solely on results in the finals.. if they win a few of those his stock goes up, but I guess he just becomes the Marv Levy of MLS? In his stint as manager they lost in the finals four times, conf. finals twice and conf. semi finals twice. Under him the Revs were 1st in the east twice, 2nd in east three times and 3rd twice.. in nine years. His last two years weren't great, but that's a pretty good 9 year run. Basically all of the Revs best years were with him as manager (other than 2013, 2014)... and that's without Landon.
I think the Revs were good in that era mostly because they had so many good players for the era. Joseph. Dempsey. Twellman. Ralston. Reis. Parkhurst. Noonan. Larentowicz. Nicol was an old-school, English "roll the ball out and play" type of manager. I've heard lots of thinly-veiled references suggesting that he spent a lot of his time golfing. I don't think it's a coincidence that he transitioned to being a mediocre pundit and never coached again.
All three managers suffered under the leadership of Mike Burns, always one of the most notorious GMs in MLS. The Revs were known for being awful at foreign player scouting and recruitment. I think this is changing a bit with Arena in charge, but New England has long been THE place in MLS where players don't want to go.
I think the success of the Revs in the 2000s and the suckage of the 2010s boils down to being both good & lucky with the draft when it really mattered, then completely failing to adjust.
In the 2000s, the draft was huge. The league didn't/couldn't attract a lot of great foreign players and there were no academies, so the draft was where you got top talent. Plus, with only 10-14 teams, there weren't as many teams competing for talents in the draft. Then what happened was that the draft got worse — more young talents bypassed it by signing abroad or by joining MLS academies, with 20+ teams you just didn't have as many good picks, and with increased investment in foreign players it became that much less likely that college players would become "stars" in the league.
Here's the Revs' drafting record. Player (round)
2002: Twellman (1), Joseph (2) (!!!!!!!) [Also picked up Steve Ralston in the allocation draft that followed the contraction of Miami Fusion and Tampa Bay Mutiny, and also Alex Pineda Chacon who was traded for Matt Reis a year later]
2003: Noonan (1)
2004: Dempsey (1), Dorman (6)
2005: Parkhurst (1), Riley (2), Larentowicz (8)
For MLS of that era, this is simply E-P-I-C. And also totally unsustainable. The Revs got basically nothing in '06 and '07, Chris Tierney in '08, Kevin Alston in '09, not much in '10 (Sinovic, but they cut him), A.J. Soares in '11. The guys I mention here are useful, but they are nothing like the absolute foundational, building block stars that Twellman, Joseph, Dempsey, and Parkhurst were.