The "other students" you speak of do not play in a situation where hundreds of millions of dollars are made from their efforts--indeed, their "ride" currently depends upon the efforts of that relative few who participate in men's football and/or basketball programs at a limited number of schools, as you note.
The idea that these young men have a "choice" is a specious one--it is only the choice of "take it or leave it", "play our way or not at all". They made THAT choice a long time before. Where else in our supposed "free enterprise" -system do we routinely accept this kind of unrestricted, ad infinitum "exploitation without redress"? Before even getting to the verge of a bigtime college scholarship they have run a gauntlet of tests and outright risks; at each level, the opportunities for serious injury ending all their dreams, wasting the efforts, long hours and hard work; now all of that, especially the risks, are about to be greatly intensified. Glory and a chance to at least vie for a professional sports career (I'm not going to get into the whole question of a college degree, true graduation rates and the "real reasons" that these so-called "student athletes" go through all of this and are at these particular 50 or 60 programs in the first place here) may seem enough to most such single-minded kids in their late-teens, but the ninety-something per cent of them who will find themselves in a minimum wage job in 5 or 6 years, AFTER the cheers and hero-worship, without a degree, many with life-long physical aches and pains, even WITHOUT counting in the specific money made on their names, likenesses and the efforts of themselves and their teammates, don't they deserve some representation and protection in return for their participation in this ongoing "operation"? The booster-cars, summer jobs and "money handshakes" (to name just a few of these) are part of the problem, NOT any kind of solution.
If you're talking about some kind of formally organized representation for these players' general interests, with resulting explicit benefits and responsibilities, who gets what for how long and what they owe in return, and so on, set down in writing in a kind of contract, I have no problem with that.
I'm not addressing here the "how?", let alone the "how much?", regarding the details of a more just approach, only stating what seems patently obvious but continues to meet great resistance nonetheless: The system is broken, grossly unfair and corrupt through and through. It must and WILL change, one way or another, and the powers-that-be would do well to get down to making wholesale changes now (and if it keeps the government from getting involved would be doing EVERYONE a favor!).