Here's how the list should look assuming equal administration support at top schools. Breaks indicate slight dropoffs.
1) LSU
2) UF/A&M
4) UGA
5) Bama
6) UT
7) Auburn
8) Mizzou
9) S. Carolina
10-13) Pick 'Em
14) Vandy
It's all about proximity to (uncontested) talent. Hiring an ignoramus or genius is immaterial to a school's spot on the list.
I think there's much more to it than just administrative support and proximity to talent. Sure, a great coach can make an average school look better, but a history of great coaches is probably more than coincidence. Alabama didn't arrive under Saban, or even Bryant. If it wasn't a great place to coach, they wouldn't have been able to lure Saban away from the NFL.
When you look at the 90-year history of the SEC, Alabama is unparalleled in terms of football success. And, really, if you want to go a step further, Tennessee is No. 2 on the list. So two teams that are middle-of-the-pack in terms of close proximity of talent rank at the top of the conference. It's probably not coincidence that those two programs have a combination of facilities, fan bases and traditions that set them apart.
On the other hand, Georgia has consistently been unable to get to the next level for most of its history, despite close proximity to a boatload of talent and relatively little in-state competition. The same goes for LSU...in terms of in-state talent alone, we agree that LSU is at the top, yet their success has been spotty over the years.
Sometimes I think proximity to talent is over-rated anyway. Tennessee made hay in the '90s because Fulmer (and Majors before him) brought in talent from Georgia, the Carolinas, Florida and even the West Coast. OTOH, the '10s have been the best decade for high school talent in the State of Tennessee in many years, maybe in forever, and UT has very little to show for it.
Sometimes proximity to talent does matter, and it's just about finally finding the right guy to lead the program (see Clemson and Dabo Swinney). For the most part, though, I think it's way more complicated. It's a combination of facilities (which falls under administrative support -- which you cited), fan bases, traditions and other intangibles. Traditions + fan bases take years to build, but become an integral part of what makes a job lucrative.