The controller has one or more burned out MOSFET output transistors which are locked into the "on" position. MOSFET transistors are great for switching large amounts of current with the downside that when they burn out they go into the "power on" position.
Although a MOSFET transistor can fail due to a manufacturing defect, it is much more likely that an overheating event has caused a MOSFET to burn out and catastrophically fail. An overheating event can be caused by using the scooter beyond its capacity due to hills that are too steep and long, sand or mud, towing, etc. Once the controller is replaced the scoter should be used under less demanding conditions to prevent failure of the new controller.
Most electric scooters made by major brands such as Razor and Currie have power switches that interrupt the power going to the controller to stop the scooter in the event of a failed controller output transistor and/or a switch in the throttle that operates a relay in the controller to stop the motor in the event of controller output transistor failure. In regards to your E300 scooter, it sounds like, along with the controller's output transistor, the throttle or power switch may have also failed. A properly functioning brake should also be able to overpower the motor to slow down and stop the scooter if the scooter goes into a runaway power-on state.
If you are not sure if your scooter has these safety features then an emergency stop switch or relay could be installed between the battery pack and controller to add an extra layer of safety to the scooter.
High mileage can cause wear to output transistors due to multiple heating and cooling cycles.