I know his tenure was much shorter, but Francisco Cordero was the best closer the team had in the 2000s and in 2 years had 3.1 bWAR. But only 2 seasons so probably HM only.
But Corey Knebel (5 years, 4.5 bWAR and part of the beginning of the electric multi-arm bullpen set up that started in 2017 through today), deserves a spot, maybe over Thornburg?
And you also have to consider Brian Shouse who had an incredible 3 year run as the LOOGY from 20006-2008, and while that type of pitcher doesn't exist today, it did for a large proportion of the 2000-2025 time frame.