Fail-safe joystick connection
Have a circuit that sits between the joystick and the controller. 

Now you can handle this a few ways.
1. Have a window comparitor on the joystick input. If it falls outside the expected voltage range. Force 0V to the controller on all motion inputs. While it may be slightly annoying for the user. You can cut the last 5% of travel for this safety if you want. When the elderly tend to have a problem they push to the extremes of motion on a controller.


Measure the supply current through the joystick. If something goes open. It will be outside the range of that current and again halt all motion controls.


Or if you have the option. Just define in your software that 0V = stop. And anything above it is valid (with some head room for your adc noise. Just remeber to also have vcc as stop. As if say just the ground wire failed on the joystick it would also go full forward speed.