It's hard to imagine that the 'university shopping' by the top players wouldn't continue even with a salary cap on NIL money. There have always been ways to launder money to kids long before NIL became a thing.
The only question once a legitimate salary cap system is in place will be how willing is a school to taking the NCAA penalties when they slip up and get caught. There is so much extra money that goes to college for winning championships there's little doubt that many schools will take that risk.