“To the best of my knowledge there are no private ambulance services ( there are ambulances for mobility but not for emergencies .”
That is not true, at least not in NYC. There are many private ambulance companies that handle emergencies.
“I heard from Hatzolah guys that if they take insurance then they have to provide care to everyone who calls them. And no matter what day of the week. And 911 may dispatch calls to their buses if they are closest to take the call.”
Even now they must respond and treat anyone who calls them. They would get in tremendous trouble if they did not. They will never be dispatched by a 911 dispatcher because they are not part of the FDNY system. That has nothing to do with being a volunteer company.