I think you misinterpreted or misunderstood the concept of public property v. private property in the IP context, or the "Oil States" framework. In the IP context, public property does not mean that it is literally a public property that literally belongs to the public.
The issue here before the SCOTUS is which branch of the government has final say in resolving IP disputes. One argument is patent is "granted" by the government and is therefore considered a public property, and thus it can be taken away by the executive branch of the government, and the government in this instant is the PTAB, which is a part of the executive branch. And conversely, if patent is considered as private property, then only the court has jurisdiction in resolving private property's dispute.
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