No sanctioning body has allowed sports to happen because the government has precluded them from happening. With no sanctioning body around, that makes insurance a major issue.
If the government was to allow sports to happen, but the schools (i.e. OHSAA) decided not to sponsor/sanction sports, other existing groups (i.e. USAW, USSF, etc.) could choose to sponsor sports. Those other groups have insurance carriers and the such, which would reduce issues with insurance. Being I am far from an expert on the insurance side of things, I don't know the full scope; but what I do know is that the sanctioning bodies tie into insurance coverage. The question would then become what type of facilities would be willing to host events based on their existing policies and relationships (schools, rec centers, private facilities, etc) - and I do agree that would be "all over the place".
Trying to keep this as non-political as possible ...
Should the government deem it safe for sports to happen (outside of the ones that are already happening), sports will happen in some way, shape or form. It may not be within the traditional "school structure", because schools don't want to deal with the politics/liability of sanctioning the activities, or they can't deal with the dissonance b/n having sports and choosing to have hybrid or fully remote educational models. Private groups may have an extended reach compared to their normal role within a calendar year to plug the hole for a public (some of whom) that wish to have sporting activity happen.