What happens when the state steps in, when a senior is in the hospital and no one has the legal rights to help with them or their affairs?

You say the state has stepped in. If the person is incompetent as judged by the courts and the medical system then the state takes over placement. If family is cooperative then they certainly can visit, take gifts, provide support once placement is done. But the state is in charge of placement, where and in what kind of facility, medical care, management of assets--that is to say of EVERYTHING.

So, worries over. It's going to be taken care of legally by the state.

Hospitals have social workers, so the senior (if incapacitated) will be referred to them. The hospital will decide if they can be safely discharged. If so, and they can get themselves back "home" (wherever that may be), they are discharged. If they seem like an unsafe discharge, social services gets involved. Depending on what is going on medically with the senior, social services will seek direct placement into a facility with appropriate level of care.

The county will acquire guardianship and then will be able to pursue finding and managing this person's assets and medical records. If they are able to contact other family members to get information, they will. Once guardianship is assigned, then family will have no more access or insights into their financial or medical affairs and decisions. Their house (or residence) will be secured, passwords will be changed, etc. Their job is to protect the senior as if it were themselves. Family will be welcomed to visit and interact in safe and healthy ways, but will have no power legally.

The state will generally get APS involved and they would find appropriate care. The state will see if any family can pursue guardianship, and if family can't or won't, the state will step in and do so.