For fully auto you could push a lot of comms through a 2.4GHz link although I'm not sure of the specifically allowed comms channels for robot combat. You'd only need to directly activate functions (motors and weapons) so you could have the laptop in the booth. So more like a computer driving it than a completely self contained robot. 3 reasons I can think of for doing it that way -
safety - you can directly intervene with the control of the robot should it be required;
simplicity - the robot could initially be identical to the human controlled one but with an AI moving the sticks;
protection - a laptop or embedded PC (or even a smart phone) likely wouldn't last 1 drop / impact and certainly not several and would be an expensive bit to replace.