I cannot remove the pot because it is a complete analog stick and I would have to desolder it out. I am not so good at esoldering and I may end up removing a solder pad or something.
I decided to give it a try and soldered the 100K potentiometer on top of the pot. The analog stick worked as it used to but the steering wheel turned at the very ends. That is, I had to rotate the steering wheel to the very end for it to work.
Maybe 100K is a big value. Should I go for 10K then?