Regarding: "Hospitals and schools can be stripped of their "Catholic" identity, sort of like the Church owns the brand. "
- Well, what really happens is that the local bishop takes the institution off his list of catholic institutions in his diocese. This means that he, himself, can not vouch for or verify the institution's catholicity and the institution's communion and union with the Archbishop of Rome.
- The institutions itself can still call itself Catholic and see itself as catholic and in union and communion with the Archbishop of Rome as the people who operate and own the institution wish it to be so.