If they are employees then they need to forfeit the scholarship and get paid a salary equivalent to the value of the scholarship on a bi-weekly basis. However, they must pay for all expenses out-of-pocket - tuition, room & board, meals, books, supplies, insurance, physical therapy, medical, etc. Travel & accommodation can be reimbursed as a business expense.
They cannot have it both ways in terms of the full ride scholarship and being paid as an employee - especially at whatever exorbitant % that is probably being sought from revenues the team generates. Does the 13th man on the team get paid the same as the star player(s)? How about walk-ons that work as hard in practice, don't play, and typically inflate the team GPA for APR purposes? Do all athletes in non-revenue generating sports get paid as well? Most football programs lose money.
Aren't they getting "paid" enough already with all the perks they get to play their sport, advertise their skills and services on national television (at least FBS football and the top 6 basketball conferences), and the full ride along with travel, accommodation, per diem on the road, school apparel, etc. in exchange for an opportunity to get a valid 4-year degree (at least still better than HS diploma) with major networking connections that can serve them in the real world when 99% figure out they'll never sniff the pros?