19. 1984 BYU
Head coach: LaVell Edwards
Record: 13-0
Author S.C. Gwynne put it this way in "The Perfect Pass," a book on the development and rise of Hal Mumme's Air Raid offense: "BYU was the very definition of an odd duck, a Mormon school in the Utah desert with a team populated by ultra-clean 24-year-olds who had done two-year mission trips abroad. They were coached by a man seemingly just this side of crazy named LaVell Edwards, whose quarterbacks threw the football more than anyone else in major college football and routinely gained vast yardage through the air. [...] The football establishment persisted in viewing BYU the way Yale and Army had looked at the Carlisle ... 70 years earlier: as purveyors of cheapjack novelty items that nobody else was interested in and that would soon disappear. It didn't help that BYU played in the Podunk-friendly WAC, with such middle-of-nowhere football schools as Hawai'i, Utah, and Wyoming, that seemed to score a lot of points on one another but did less well against national opponents. It was another reason to ignore the Mormons and keep them off the airwaves. That didn't alter the fact that BYU scared the he-- out of everybody."
That team won a national title in 1984. The Cougars benefited from a particularly chaotic national landscape, with every top team losing at the worst possible time. But they still did it. It might forever be the most establishment-unfriendly result that this establishment-dominated sport will produce. That gets you on this list.
https://www.espn.com/college-football/insider/story/_/id/40321386/college-football-bill-connelly-most-influential-teams