There's a bit of a chicken or the egg thing going on.
I would argue that they get whatever they need from the athletic department because Saban has proven that investment will be worthwhile; they aren't successful just because they get unlimited support from the admin. In Saban's earlier years, he was successful just because he's an incredible coach. The admin over time recognizes this and begins to give him virtually unlimited support.
Saban would never admit this himself because it is against his ethos of never being complacent, but the program runs like a machine now. You are totally right about it being run like an NFL team. I don't even think they really have to recruit anymore. All the top players just want to come to Alabama because they know they'll be in the NFL in 3 years, and they can pick and choose from among all the 5-stars the best ones for their system. But this is the case because Saban built the machine during his first 3 years there.
From a motivation perspective, the thing I give Saban the most credit for is not letting guys rest on their laurels. If you are a highly touted high school player or have already won a title at Alabama, how easy would it be for you to think you don't have to work to continue to win? We see other teams get complacent after a single big win, much less conference or national titles, yet Saban never seems to have guys with those attitudes on his teams. Or if they do have that attitude, they definitely don't play like it. I don't quite know how he does it.