I think you're right, that's what the cut & carry goat people are doing here. Huge tractor, huge wagon, plain disc mower on the front, they absolutely fly around the paddocks and get through alarming amounts of area in a very short amount of time.Wagon and front plain disc mower is probably way to go
Minimal damage to the grass from what I've heard (apparently goats are quite picky when it comes to bruised grass?)