If I built a super-scaled-down and dumbed-down replica of a liquid fuel engine, do you think it would still function relatively the same?

If you could build each piece properly then I would work. There many very small liquid fuel engines used in model planes. Most of them are two cycle so it eliminates the cam and valves. They also use a glow plug rather than a spark plug. In other words they are closer to a two cycle diesel engine.

Depends on what systems you eliminate and "dumb down"----When you describe a liquid fuel engine, do you include details that make it practical---work in the real world?   The design of the pump impellers, the circulation of coolant within the Nozzle to keep it from melting?, etc.-----These are practical considerations that must be addressed for the engine to actually WORK--but are not usually gone over when you are simply explaining HOW the engine works.-------I likely will NOT work in the real world because of the small details secondary to the basic principle of the engine;s operation would not be implemented.

Go to an RC Aircraft hobby shop and look at the variety of engines they have.  2 cycle and 4 cycle engines.

things like that don't scale well due to the various laws that don't scale.

for example: A chamber half the size has 1/8 the volume, holds 1/8 the thermal energy, and can transfer 1/4 the heat (due to surface area).

If it can burn fuel and turns then it will work