and they switch Engine 1 off before entering stand so they can get the air bridge or in EasyJets favourite case the Stairs
You haven't got that bit quite right.
Nobody and nothing should approach the aircraft until the red beacon/anti collision light is off, the exception is the GPU guy. As both engines are turned off within seconds of each other, if you've left them both running, you aren't really saving any time by shutting one down.
The single engine taxi is for fuel saving purposes. You usually do it when you judge that you can reach the stand without needing to stop or use too much power. If you do stop too early, as the case appears to be in your second post, then it needs a pretty big handful of power to get the thing moving again.