I have to say that I'm at a bit of a loss with this story....
Firstly, the 737 referred to by the OP is in a clip being used to demonstrate what currently happens; it's not specifically relevant to the 'new' system.
Secondly, the tug does still have a driver on board, so presumably s/he would just drive the tug back to the terminal - or to some other point where aircraft are to be towed from.
However, and this is where I lose the plot slightly (and others posting here would be far better-placed to comment than I), I don't understand quite how this would work in practice, and why it makes a jot of difference to the engines being on whether the pilot is in charge of the tug operation, or the tug driver.
For the most part (queueing excepted), surely the engines need to be on to complete all the pre-flight checks, many of which relate to power and fuel systems.
If I'm mistaken in that, then what would the implications be, at somewhere like LHR or JFK, for how many extra tugs (in themselves no cheap things, I assume) you'd need, and how would they move around the airfield once they've deposited their aircraft at the threshold?
I do understand the huge cost of fuel burnt during queueing, but surely if tugging of this type was feasible as an alternative, then tugs would already be used in such circumstances, even if the tug driver was in control rather than the pilot.
It just seems too simple to me....