Regarding running QBs - my feeling is that a QB is there primarily to pass the ball, but a good passer who can run adds a lot of value. The top mobile QBs (mainly thinking Jackson and Allen here) are good enough passers to be effective if they just played from the pocket, but the threat of QB runs and scrambles forces the defence to play differently than they would for a pure pocket QB like Brady or Manning (e.g. needing to assign a defender to 'spy' the QB, or having the defensive line be much more disciplined when rushing so that there aren't any holes for the QB to escape through), which opens up more opportunities for the passing game. There's also value in having a QB who can escape pressure to give himself another opportunity to make a play when a less mobile guy will have been sacked (although this doesn't have to be through sheer athleticism - just being hard to tackle is sometimes enough, as Ben Roethlisberger showed many times).
A mobile QB does need to be at least reasonable at passing from the pocket, though - if not, the defence can do the bare mininum in pass defence, devote almost everything to stopping the run and dare the QB to make plays (see the Colts and Anthony Richardson for a good example of this).
Sneakylong Actually Bill spent most of the Brady home discount money on the defense. They did have a pretty good offensive line over the years that protected Brady though. I’m a traditional drop back QB guy myself.
While the Brady-era Patriots did invest to some extent in their offensive line, they were also very good at finding quality starters in the late rounds of the draft and free agency, which freed up resources for the defensive side of the ball. A lot of the credit for that has to go to their long-time offensive line coach Dante Scarnecchia - there was a noticeable drop-off in the performance of the OL in the years that he wasn't there.