Like many services in the US, it is highly dependent on the region. In the places I’ve lived (in California and Washington state), each county generally contracted with a private ambulance company (or sometimes several, just to be confusing) to provide transport services. Even if an individual city’s fire department staffs their own ambulances, they may still attempt to bill insurance (since they’d be leaving money on the table at the expense of their taxpayers otherwise). Some of them may cover city residents free of charge but bill people from out of town. It just all really depends.

I would definitely dispute your last sentence — in many places, ambulances are absolutely part of the predatory American healthcare industry. Plenty of people will try to avoid calling an ambulance or try to find an alternate ride to the hospital, since they know an ambulance ride may end up costing them thousands of dollars.