That is simply not true. If a police officer responds to the scene of a possible break-in and finds a person in the house that lives there, he/she would be remiss if they didn't also verify that the person who is rightfully in the home is actually the person who was observed breaking in.
If the police had left and Gates was attacked shortly thereafter by an intruder who had already been hiding in the house when the police arrived, we'd never hear the end of it . . . and that professional malcontent Looie Gates would be making a racial issue out of the lack of proper diligence by the white police officers who came to the house.
Oooohhh.... A duty to whom? The homeowner? If the homeowner demonstrates he is the person who rightfullly lives there he doesn't have to justify the presence of anyone else who happens to be there to the police. The police can ask 'do you mind if I check the rest of your house?' but if the homeowner says 'I would mind' you're telling me the policeman can do that without a warrant? I don't think so. I really don't.
You can apply your scenario to any situation when the police turn up. 'Oh, we had to search his basement to make sure there were no armed intruders down there. That's when we found all this Nazi memorabilia and he began yelling and we had to arrest him...' It never stops.