Safe havens came into existance so that women could drop off unwanted babies with no questions asked. Demanding them to give medical information or to identify themselves is contrary to what safe havens were created for.
There's a reason women use safe havens. They don't want the loaf and they want to get rid of it safely and anonymously. It's why there use to be closed adoptions. I couldn't imagine dumping off an unwanted baby and having people demand my personal and medical information.
At that point, it's better to go into a remote woods, dig a deep hole, give birth over it, and fill it in afterwards.