Chosen Solution

I was finished changing Pro Controller shells until I found the right stick feeling crunchy when I clicked it. After clicking, the stick would often get stuck on the right side, and when I opened the controller again, I found this part popping out when the stick was pressed. If this is part of the problem, how can I go about fixing it?

Replace the whole joystick module. Desoldering it is kind of tricky with soldering iron alone. Way too many soldering holes to clean up at the same time, a few of them are even connected to the beefy joystick frame, sometimes the legs are twisted to hold it in place in the hole. Using a hot air rework station to blow from the back probably helps heating it up evenly, but you have to make sure there is no other things in the back that may be damaged by hot air.