I wouldn’t flush the coolant, adds potential for an air lock, making things confusing to diagnose.
Having new coolant won’t affect the temperature. The additives in coolant are mostly to prevent corrosion and freezing. You could fill one car with water and another with coolant, they would run at the same temperature, it’s the pressure in the system that prevents the coolant/water boiling and turning to steam.
Don’t underestimate the coolant filler cap, it works as a valve, allowing air (preventing a vacuum) into the cooling system as the engine cools, it also works as a pressure relief valve allowing excess pressure in the cooling system to be vented out, otherwise the excess pressure would burst hoses.