Right "bumper button" does not function when pressed?

To answer your question:

1. If you've turned the controller over it has a weird proprietary screw that Nintendo used for the SNES, N64, and Gamecube. Its commonly referred to as a Gamebit. The size in the controller is 3.8mm. The size in the console itself is 4.5mm. Be wary though, every time I've gotten a Gamebit set it either doesn't fit properly or is made with inferior metal and tears itself up quickly.
2. The button feels ok, so the spring for the analog depression of the button should be fine, but you can try cleaning out the inside in cases of excessive dust or grime. But unfortunately its probably something more insidious, along the lines of improper communication with the console. After a 24-bit command word from the console, the controller responds with a string of bits that contain the state of all the buttons along with joystick position data. The 8-bit R button sequence is at the tail end of that response(at least the analog portion). The digital signal should be in the middle of the button state data, so if no digital signal is happening it could also not be properly connected or communicating with the unit that updates its status to the console proper.

SUMMATION: try opening up the controller and cleaning it out, possibly also realigning things if they're out of order. If that doesn't work, you may need to get a new controller or an electrical engineering degree.

For you! (Nintendo branded Gamecube Controller)

Of Note: The red circles are parts corresponding to the analog input, they should freely move. The yellow squares are the contacts for the 'click' of the digital input, they should be clean and properly aligned with the carbon impregnated pads at the end of the shoulder springs. Hope this helps.