By definition, a non-profit does not have shareholders, so while an ESOP may not "show a profit", it's because, like an S-Corp, those profits are "distributed" to the shareholders. Where they avoid taxability is by deferring them to retirement funds.
And frankly, that may be one way to allow direct payments to players and NOT affect Title IX, inequalities between athletic and non-athletic scholarships, etc. Some sort of ESOP or LLC set-up, where deferring or redistributing the payments and profits to educational, or even post-grad educational funding. Or, like any employee enrolled in retirement or pension plans, they leave school, they roll that fund to next employer.
Where I (personally) see the biggest problem, and thus Swarbrick's most valid point, is not NIL, or Transfer Portal, or whatever, but rather, the confluence. As he is quoted, "We went from what some called the most restrictive system to the most unrestricted free-agent labor market in the history of all sports".