First of all, this isnt' really Florida asserting its marks. It is Collegiate Licensing Corporation (CLC), which is the licensing and enforcement arm of about 98% of Div. 1 schools out there. Now, Florida could certainly step in and put a stop to it, and conversely Florida might have asked CLC to ramp up the enforcement, but its primarily out of Florida's hands.
Second, while I am sure CLC is willing to work out a reasonable trademark license for the High Schools, there are some problems. CLC cannot just enter into a cheap-*** sham license, or they lose control of the mark. Letting a High School use the UF logo is essentially the same as putting a UF "Seal of Approval" on everything the logo touches. They have to set certain standards for T-shirts and the like, and there could even be a conflict with Nike's license to use the logo on jerseys and things. As you can see, a license like this would not be cheap, and most High Schools are on pretty marginal budgets as it is....