Lets explore this:
Does/is the cooling system designed to run with the expansion tank cap on, well naturally.
However while diagnosing issues we often have run the car for some time with the expansion tank cap off.
There are some caveats:
Since the cap is off the rate of expansion of the coolant will be greater, therefore expect some spill out as the cooling system self levels, if you will.
If the fan(s) be it electric, mechanical and/or both are operating properly the temperatures will stay within parameters.
The coolant temperature sensor for the DME to make decisions about opening up the thermostat is located (not by accident) in the cylinder head, when the temperature rises above XX°C the DME will send a PWM signal to the thermostat to "drive it open".
Having the cap on or off in no way changes the flow and/or path of the coolant.
Would I drive a car with the cap off? No way. Albeit at idle engine speeds once the level stabilizes (pressure is absent) one can run a car for diagnosis at idle, provided you started with a cold/warm engine.
All of the above becomes pernicious and not recommended for the layperson. Should there be an issue (poor pump flow, bad fans/fan clutch, bad thermostat going too hot) one can easily get scalded and a trip to the hospital is in order.
However the cap on/or off has no effect on flow capability, as long as the engine itself remains full.