Before getting a new controller, it would be nice if we could eliminate the throttle as beeing a culprit. You could get the same symptoms if you had a short between the red and white throttle wires. It's unlikely though that that's your problem because most controllers check for it when you switch them on, and they won't give power if they detect it Could you unplug the white connector on the pcb and see if it still immediately goes when you switch on. What do those two green wires do on the same connector?
It's a pretty good bet that you have blown mosfets because it's a very common problem with brushed controllers. If you have a meter that can test continuity (beep or resistance), you could confirm it by measuring between the positive battery wire and the positive motor wire, and then between the two negatives. They should be open, i.e. no beep or infinite resistance. If it beeps or you see zero resistance, one or more mosfets are blown.
The battery wires are connected direcly to the battery wires, but the current is blocked by healthy mosfets. When you open the throttle, the circuitry makes the mosfets pulse open to let the power through like a sort of valve. When blown, they're open aĺ the time, which is why you get maximum power.
The watts rating of the motor is pretty meaningless. Mostly, they rate them low enough to comply with regulations. The Heinzmann motor is rated at 200w, but the controller allows 28 amps at 36v, which is 1000w. I believe that the old Powabykes run at about 20 amps, and looking at your controller, I can see at least four mosfets, wich implies about 20 amps. The one I linked to was 21A IIRC. If you notice that it gives more power, you can always use a bit less throttle. Motors don't burn unless you give too much power when going slow climbing up steep hills. Sometimes, more power makes them climb better, so you get less overheating.