I don't know if this kind of move would shield Rutgers Athletics from NJ's open records act. So I don't know if that is a benefit.
The clear benefits to Rutgers would be to separate Rutgers Athletics finances from University Finances. But as long as the University continues to subsidize Athletics, I don't think this move makes sense. I'm not sure how the subsidy would work under this type of structure. The structure really only makes sense for self-sufficient athletic departments.
As far as control, the FSU example puts the Athletic department corporation under a board consisting of the University President, Athletic Director, a member of the university governing board, a faculty representative, and the head of the athletic fundraising organization (at Rutgers, this would likely be equivalent to the head of R Fund).
So how is this different from Rutgers from an oversight perspective. At Rutgers the athletic department is already run by the Athletic Director who reports to the President and there is oversight from a member of the BOG who chairs the athletic committee. So that part is pretty much the same. Rutgers doesn't have direct Faculty involvement, so that part changes. Also, at Rutgers the head of R Fund reports to the Athletic Director with dotted line reporting to the Rutgers University Foundation. If Rutgers reorganized similar to FSU, the relationship between R Fund and RUF would change, and R Fund would probably need its own board of overseers.
So the only real change would be more financial autonomy for Athletics, but that doesn't seem to be a benefit if Athletics is being subsidized, as is planned for the foreseeable future.