Its common in the US for basic life support to be delivered by private or hospital ambulances, or municipal EMS services by cheap EMTs. Sometimes they are just drivers. Usually advanced life support is done by paramedics or others who work for municipal EMS or fire departments.

Bigger paid departments have a split between engine companies and rescue companies. Rescue companies have fancy tools for all sorts of situations and usually have more trained medical capability. That’s why you’ll see ladder trucks dispatched to medical emergencies in the US.

All right then, “most communities in the US.” I’ll admit I don’t really have any idea how it works anywhere else.

However, I can only imagine that all developed countries have the “problem“ that modern building codes reduce the rate of fires such that they are faced with the choice between (1) reducing density of firefighters and therefore increasing their average response time (2) paying them to sit around and do nothing, or (3) giving them something else to do when there isn’t a fire.

In the US it is more likely that there is a fire department close to any particular residence (in fact they are usually spaced out to minimize response time over the entire area) while ambulances tend to be dispatched from hospitals or ambulance services and so their response time is more varied. Fire department paramedics and ambulances may both be dispatched for emergencies, but ambulances also cover things like non-emergency patient transfers and other medical transport scenarios. It is often the case that the role of the FD paramedic is to be first on scene and try to stabilize the patient if possible so that once the ambulance arrives they can scoop and go if needed.

It's that, be run by the police, or be a completely separate institution. Private healthcare makes hospitals in control a bad idea.

Fireman are for protecting life, police for protecting law.