You can have a spending cap for teams while also allowing athletes to have NIL outside of the salary.
Just like Pat Mahomes for example has a salary that applies to the Chiefs' salary cap, but he also makes income from State Farm commercials and other appearances.
Once athletes are school employees they officially become public employees and their salaries would have to be public record.
There wouldn't really be "under the table" and teams won't exceed their caps, because they'd just pay the baseline and let collectives make up the rest in cooperation with the athlete.
This could also tie in some 1-2yr contract deals with the school. Where players are tied into the team for X number of years before being a "free agent." The first contract could be exclusive rights like it is for rookies. Say 2 years with a 3rd year option if a redshirt is used? Everything after is 1yr contracts. If a player transfers before the end of their contract the new team buys out the contract.