Jet engines will run on pretty much anything. You can burn unleaded pump gas. I always joke you could burn powdered coal, if you could find a way to get it in the combustion chamber. You could burn leaded gas in an emergency, but you were time-limited because of the lead build up on the turbine blades.
The B-36 fed its jet engines from the same fuel tanks as its 6 piston engines.
The real reason there are so many fuel is the other properties. Avgas is more volatile, and you can get (effectively) vapor lock at altitudes. Plus the whole explody-thing. Jet fuel is less volatile (though watching jet airliner crashes, with the huge fireball, you might wonder. But generally, if you drop a lit match in jet fuel, the match will go out. It’s basically diesel.).
JP-5 is even less volatile, which is why the Navy used it.
The fuel differences make their appearance at extreme conditions. High altitude, cold temperatures, humidity, all affect how the fuel behaves in the aircraft. With some jet fuels, too much water and never draining you fuel tanks can give you what is known as “apple jelly”, a karo-syrup like buildup than can clog your fuel system.
Also, the choice of elastomers for O-rings and other seals depends on which fuel you are using. These days, it’s not as critical, but in the past some elastomers reacted poorly with different fuels.
In brief, you can use pretty much anything in the short term. Long term might cause breakdowns of the fuel system components.