A couple of things about this -
First, having paid employees is not the problem. Hospitals, churches, schools - all have employees. The difference here is that the employees are being paid to advance the exempt purpose of the not-for-profit. A football player is not.
Second, it doesn't have to be a separate entity to be taxable. Tax exempt organizations have something called UBIT (Unrelated Business Income Tax). Many, if not all, complex exempt organizations have to deal with this tax all the time. One of the most common issues is advertising - the Football Program, for example, probably already creates UBIT issues because the sale of advertising is not related to the exempt purpose. This article explains it pretty well. The tax rate is a flat 21% on the net income generated by the unrelated activity.
https://www.aicpa.org/resources/article/unrelated-business-income-taxes-ubit-in-a-nutshell
So, why would paying employees trigger the loss of exempt status? Because these athletic programs were already on thin ice over a decade a go. Athletes receiving direct compensation instead of playing football as a part of their commitment to the school for academic reasons. would like be enough to push it over the cliff.
And finally, employee players "might" have to pay taxes? It's a mortal lock that they will. Unless it's a small enough amount that they can consider it part of the scholarship (total cost of attendance).