have aircraft ever been pushed back & towed to the holding point & started engines 2-3 minutes prior to takeoff
Only the first one. All the others get delayed by the tugs clogging up the taxiways returning to the aprons.
Plus, what if you have start problems? Do you want to find out close to the apron where you can return for engineering, or 100's, maybe 1000's, of metres away at a holding point, blocking access for everyone else.