Thanks for the information. I guess I was mixed up because of how church property and state is managed in the U.S.
I believe the only thing we have that’s remotely close to a cathedral like Notre Dame is Washington National Cathedral in DC and it’s 100% privately owned and operated by the Episcopal Diocese of Washington. I’m pretty sure that even includes state funerals that happen there.
I know Florida has a church that is older than the Washington National Cathedral. San Antonio has a San Fernando and the Missions that are certainly older too. The Missions are a national park, world Heritage site, and active parishes. San Fernando is certainly though owned by the Catholic Church.