The problem is GSX provides more than just pushback services. In addition to pushback, you also get boarding, de-boarding, catering, and refueling, so if you did any of those with the engines running, bad things would happen. GSX has no idea which one you're going to pick. As far as starting engines before pushback is concerned, in most cases, you wouldn't do this anyway. The only time you really would is if your APU was inoperative and you needed the ground air cart for the engine start, then you'd start at least one at the gate.
