Seems to me there are several factors at play here.
1) Cost. Is there a significant difference in rental fee?
2) Pitch quality. I suspect Stone is "better."
3) Stadium capacity.
4) Spectator safety. Seating on both sides, with fans separated, not the worst thing.
5) Geographic fairness. Should there be a rotation?
6) Recruiting fairness. If one university hosts, will others want to, in order to gain some perceived recruiting edge?