I don't see that as a reasonable constraint - nuclear propulsion does not need combustion, thermo electric does not need combustion, and there's also things like cold gas thrusters, laser ablation etc. etc. etc.
Usually those are called thrusters rather than rocket engines.
Hydrazine or hydrogen peroxide can be used as monopropellant, and it's still rocket engines, even without combustion.
True, but it's still a highly exothermic decomposition, the energy comes from a chemical reaction, unlike ARCA's silliness.
But it's not combustion. I guess you meant something else. Now you label they approach; do you think laying boundary line this way looks obviously correct? Say, nuclear rocket engines heat up gases - also without chemical reaction; resistor jets also have reasonable history in rocketry - why ARCA's approach is wrong?
Because their vehicle ISP is ~90-100. Plug that number into the rocket equation and see how much delta-v you get.
There's no combustion requirement for a reaction drive ;-)