To add to this (and agree with you), the Bills ran more often than they passed in three of their four Super Bowl seasons, and five of six years during the period of AFC East dominance which started in 1988.
1988: 528 runs (Thomas 207, Riddick 111, Mueller 81), 454 passes
1989: 532 runs (TT 298, Kinnebrew 131), 478 passes
1990: 479 runs (TT 271), 425 passes
1991: 505 runs (TT 288, Davis 129), 516 passes
1992: 549 runs (TT 312, KD 139), 509 passes
1993: 550 runs (TT 335, KD 109), 497 passes
Some of that can be attributed to running out the clock at the end of wins ... but when a team's primary back is at/above 300 carries, "pass-happy" might not be an accurate description. To compare, Marshall Faulk had 324 carries in his final season in Indy, and never more than 260 during his years as a Ram.