Which will probably be about an hour into the cruise on a 777.
Also, you have to factor in some engines have a minimum warm up time. My current type, it's five minutes. Not only are the engines going to have to be started while motoring/being towed, there's a big risk of missing a slot should either the crew get the timing wrong or an earlier slot become available. This will inevitably lead to some delays. Plus also consider the hassle of a potential start fault away from the ramp.
Monarch used to have a DC10. To save costs, they used to start the number two just before line up. A great idea, except that they were then getting airborne without it having warmed up sufficiently and they had a couple of in flight shutdowns that engine. A number two engine change on that type is a huge problem. It cost them far more in the long run.