I agree with your points - free agency is one of the tools a good GM uses in the off-season to help improve a roster.
I think 2001 showed that Belichick could use free agency effectively to help build a team - it's the gold standard that other GMs look at as how money can be well-spent during the free agency period.
Mike Vrabel, Antowain Smith, Bryan Cox, Roman Phifer, Larry Izzo, Anthony Pleasant, Mark Edwards and David Patten were all key pieces of that team that were added via free agency. (Otis Smith and Joe Andruzzi were free agents Belichick added the previous year.)
Hitting on both Richard Seymour and Matt Light in the 2001 draft was also a key. And I think there was a guy from the year 2000 draft that came off the bench and did his job.