The usual procedure for flushing a cooling system is to drop the existing coolant (usually by disconnecting the bottom radiator hose) fill the system with distilled water and whatever flushing chemical the workshop uses, restart the engine, drive it around for a short time until everything is at operating temperature, dump the water and flush mixture and refill the cooling system with the proper coolant.

You can also simply flush the radiator (and not the rest of the cooling system) by disconnecting the radiator hoses and running a hose through the radiator, usually in the opposite direction the coolant flows. But this is a much more simplistic flush and doesn’t change the condition of the engine crankcase’s cooling passages.

It’s possible that the flushing process has loosened some rust or scale that then blocked the cooling system somewhere else (the thermostat is a prime suspect) and if that’s the case, the process needs to be repeated until the water coming out of the cooling system is nice and clean and not tainted with rust or scale or anything else.

Revving the engine to get the flushing mixture to do its job is not really a problem, but could also be achieved by actually driving the vehicle (which would also get it up to temperature faster). But, really, provided the temperature gauge hasn’t been allowed to move into the red zone, you shouldn’t have damaged anything. But you do need to know why the operating temperature of the engine is now higher after a cooling system flush.