A Wifi plug should be able to allow power flow while "on" and stop power from the outlet when "off", it would have the same effect as unplugging / plugging the unit back in.
It does not help. You need to press the button.
My mistake, not the best feature to have a button during large scale deployments if you want to be able to remotely cycle units.
Has anyone checked the behaviour if you tape down that button? Then additionally during a mains power cycle? I would but I'm away from my unit at the moment.