He has already proven that he can turn a program around (Rutgers) and he has coached in the NFL.
I don't think big time programs are sold on Coach Schiano as proven head coach at the Power Five level, and here's why. The 'rise' of Rutgers under Schiano (HC from 2001 to 2011) coincided with the 'decline' of the Big East.
In Coach Schiano's first four seasons, Rutgers won a total of 12 games. His first winning season (7-5) was 2005. In '06 they went 11-2, came in 2nd the Big East, went to the Insight Bowl and finished #12 in the AP Poll. That was the high water mark. From 2007-2009, Rutgers never won more than 9 games, and had a 4-8 season in 2010.
What was more significant in the Big East at the time was the teams that the ACC poached from the conference. Va Tech and Miami left in 2004. BC was right behind them in 2005. Then Rutgers goes 11-2 in 2006 against a slate of UNC, Illinois, Ohio, Howard, USF, Navy, Pitt, UConn, Louisville (final ranking #6), UC, Syracuse and WVU (final ranking #10). The Scarlet Knights beat Louisville and lost to WVU. Essentially, Rutgers was playing in the American Athletic Conference before it became the AAC. The years they won 8-11 games, they weren't playing Miami, Va Tech and BC.
I'm not saying that Coach Schiano didn't vastly improve the quality of football at Rutgers; he did. But, it's not like he turned them into a perennial Top 25 team, or conference championship team. They were ranked at the end of one season, and they never won their conference.
I think he's a great coach. But, I think his jump to the NFL was premature, and I think he needs to prove himself at a middle tier program before a blue blood or major Power Five school will take a chance on him.