It's probably not an easy job do, as I found out myself when having to replace a poly-800 joystick.I used the original pots. The potentiometers have a centre position that I think bypasses the pot. If the stick isn't centreing properly it can cause drifting.The pots themselves are very good quality and would probably just need a clean. Scratchiness shouldn't be much of a problem as the pot doesn't really effect the audio it's just a control signal I believe.
The switch on the playstation style stick you linked could be used to bypass the pots so that if you want joystick control you would press down and hold but do the pots on it go 10k-0k-10k or just 0k-5k-0k?