Yes if I had to pick a conference to raise the homer flag, it would be the BIG. But the SEC has won a NC 3x since the last BIG title.
BIG has a long 17 year drought between titles, not the SEC. Pac12 has an even longer drought. Big 12 is too Kansas heavy. The Big 8 - 12 hasn't crowned a NC outside KU since 1945-1946, when OSU was OA&M.
As for the BIG, plenty of opportunities to win a NC the past 20 years (5 runners up: MSU, Il, OSU, IU, & Mi), far more than the Pac12, Big12 and even Big East. At the end of the day, BIG lets too many top shelf recruits leave the footprint who go on a win a NC elsewhere. BIG now has to deal with a greater Big East presence. Both conferences share the footprint in 6 states: Neb, Wis, Ohio, NJ, PA, & In & Georgetown is roughly 15 miles from College Park, MD.