Running engine to temp for winterization

Do you need to run the engine up to temp so the thermostat opens before putting in antifreeze? All I did was remove all of the drain plugs, drain all of the water out, reinstall plugs, run antifreeze through until it came out the exhaust. So did any antifreeze get into the engine? Also, if you need to run it up to temp before putting in the antifreeze how much time do you have before the thermostat will close? You would need some time to shut off the motor and remove all the plugs to drain re-install and start putting in antifreeze. Is this overthinking?