You think Hoover High School in Alabama is paying $125,000 to their head coach straight from the school? And none of that money is being given by boosters so that Hoover High School football remains Hoover High School football? Your claims are fairly ridiculous about where this money comes from. No school system can pay their coach that much money out right, the "coaching" supplement comes from boosters. You think the many many football coaches in Texas are paid strictly from the school systems? No, its the booster who pay those supplements. The rich oil industry guys who want their high schools to be the best on the field on Friday nights give the money to raise salaries. You know all about Alabama's rank in the national education spotlight. Where is Texas in the national rankings? I would assume ahead of Tennessee, both in academics and athletics.
Also, it is not illegal for a booster club to pay coaches. It varies by school district, and I also believe that the TSSAA policy in regards to this is that any money given as a "supplement" must be noted with TSSAA. So, I take that to mean that it is okay as long as you have let it be know to the states governing body for high school athletics.
As far as coaches magically getting smarter by being paid more, they don't. You fire the subpar coach and hire a better one. Also, when a coach doesn't have to teach, they have 8 hours a day to plan for practice, game plan, administrative work, etc. But when you have a higher pay grade, your schools football coaching job gets a whole lot more appealing to better candidates.