QB's drafted by BB:
2000, Round 6, Pick 199: Tom Brady, Michigan
2002, Round 4, Pick 117: Rohan Davey, LSU
2003, Round 6, Pick 201: Kliff Kingsbury, Texas Tech
2005, Round 7, Pick 205: Matt Cassel, USC
2008, Round 3, Pick 94: Kevin O'Connell, San Diego State
2010, Round 7, Pick 250: Zac Robinson, Oklahoma State
2011, Round 3, Pick 74: Ryan Mallett, Arkansas
2014, Round 2, Pick 64: Jimmy Garoppolo, Eastern Illinois
2016, Round 3, Pick 91: Jacoby Brissett, North Carolina State
2018, Round 7, Pick 219: Danny Etling, LSU
2019, Round 4, Pick 133: Jarrett Stidham, Auburn
On the surface, that looks like a mixed bag, but for many, many years they weren't even looking to draft a starter. They had Brady almost that entire time and were usually only in the QB market for a backup. They got more than they invested in Cassel (who made the playoffs and a Pro Bowl with KC). Garoppolo looked the part in NE and has done well in SF so far. Brissett ended up starting in IND (although that certainly was not the Colts plan).
Total NFL regular season starts:
Brady - 283
Cassel - 81
Brissett - 32
Garoppolo - 26
Mallett - 8
In that time, first round picks used on QB by team:
4 - CLE, WAS
3 - BUF, DEN, JAX, NYJ, TEN, BAL
2 - ATL, ARI, CHI, DET, HOU, MIN, NYG, LAR, WAS
1 - CAR, CIN, GB, IND, KC, LAC, MIA, PHI, PIT, OAK, SF
0 - DAL, NE, NO, SEA
Yeah, I get it. They didn't really need a capable QB because they had Brady. But even though that drafting record looks spotty (and they used 0 first round picks), here is a list by team of all QBs drafted in that time that went on to start 25 or more NFL regular season games:
4 - BAL, JAX, NE, NYJ
3 - ATL, BUF, CHI, CLE, DEN, LAR, MIN, WAS
2 - CIN, DAL, DET, HOU, LAC, MIA, OAK, PHI, SF, TB, TEN
1 - ARI, CAR, GB, IND, KC, NYG, NO, PIT, SEA
Twenty-five games doesn't sound like that high a threshold . . . but that is really hard to get to in terms of being an NFL starting QB. Not even being in the market for a QB, the Patriots tied for the most QB's to get 25 starts. So I would count DRAFTING QBs as a decent strength for New England.