I just finished putting the wheel back together. Thanks for the advice!
As it turns out, the button itself wasn't the cause. There is a rubber "padding" between the button and the.... "electronic plate" (sorry I don't know the word for it, I hope you know what I mean
). That rubber part was worn out. Luckily there is an identical part used for the buttons on the other side of the wheel, so I just replaced them. As a result my POV is now a bit sluggish, but the others are fine. I guess you could call that a problem.. half-solution?
Also, that GT logo is put on using double sided tape. As a result I completely destroyed it when taking it off. Luckily, I have an old DFP lying around, so I took off the badge from the other one (doing a better job this time) and replaced it. Now that I have the old DFP out though, I plan to somehow make my old pedals usable again (Logitech asked me to cut off the plugs and send it to them, so that's why I need to work on it).
I figure: the pedals have two separate axis, so given that I get my hands across on an old joystick (which also has 2 axis) maybe I'll be able to work something out? If anyone's interested, I'll keep posting if I get any progress!