EMR here so I have some insight on this.

As far as why firetrucks respond to medical emergencies, it's because they're also trained EMRs (previously called first responders) so they can provide some treatment on the scene even if they can't provide transport or do all the things an EMT or paramedic can do. In many emergency situations fire crews are also able to act where medical responders couldn't. In a car accident they can cut up the car so the patient can be removed without disturbing a potentially broken spine. In confined spaces they have SCBA gear to enter safely where normal medical responders can't do anything. Hell, if you wanted to, you could extract a patient through a second story wall with the equipment firefighters have at their disposal. There are many times where firefighters have resources that the ambulance crew don't have access to. Firetrucks, due to the fact that they are built like tanks, are also used for trafic control. If a road needs to be shut down they just park a fire truck across it. If there's an accident along a busy road they can park the fire truck between the responders and traffic so any lookielou hits the fire truck instead of the ambulance crew which is a frequent enough occurance that it is specifically covered in training. There is no better friend to EMS than a firetruck on scene.

As far as why ambulances are so huge, it's partly because they are mobile emergency rooms. They have everything you could possibly want to stabilize a patient in them plus spares and enough room to work around that patient while on the move. The other reason they're so big is because they're designed to be hit. While they aren't as indestructable as a fire truck, they are designed to come out as the winner in most car accidents because the likelihood of them being in a collision is much higher than most vehicles and they frequently carry unsecured people in the back. Ambulance accidents are also something we were specifically trained for. If it does happen then the ambulance may not always be able to take a hit and keep moving like a firetruck could, but it will almost always keep those in the back alive if not entirely unharmed.

Sorry for any spelling mistakes, that was more of a wall of text than I planned and I just woke up so my brain isn't working yet.

In Europe, they treat patients at the scene, whereas the US extracts the patient and transports to nearest hospital (and stabilize inside the ambulance). These approaches require very different types of equipment and manpower.

One reason (at least) with two parts: US cars are too large, and fire departments in the US cover massive areas.

American care are huge, so fire services rely on their trucks to clear obstructions in an accident. In asia/europe, manual clearance is still the norm, but you just can't do that with ex: a rolled F350 that's blocking stretcher access down a two lane road. And given how remote a call might be, every second matters because its 25 minutes by air to the nearest hospital.

The other aspect is that huge coverage areas mean any given call could be in places of deep urbanization or the most remote rural bastions of podunkia. The trucks need to be able to go offroad, carry enough material to address emergencies entirely unsupported (big water tanks) and bash their way down a glorified ATV track. Dispatch areas are too large to have specialized FDs, so city roads have to be big enough to support trucks that can do both.

In short, we have to have big trucks because we won't fund enough fire departments to allow them to specialize to a specific region's geography.