Some controllers have over Voltage shutoff circuits which prevent them from being overvolted. Most controllers do not have this feature though. The motor will almost certainly not have an overvoltage shutdown feature though.
If you over-Volted and over-Watted a 48 Volt 500 Watt motor to 60 Volts 1000 Watts I would not expect for it to last very long due to overheating.
I have seen OEM TaoTao ATE 501 controllers and the headlights, taillights, and turn signals are not wired through them and are instead wired through the scooter's wiring harness. If your ATE 501 has the lighting wired through the controller then that would be a very difficult controller to replace and unfortunately, we would not have a solution for that.