Is there supposed to be a black coating on the two tips
of the rubber button inside (touching the PC board contact
area when pressed?). I noticed that this button is ~ white
while most other buttons (that work well) have a black tip
inside...

Probably just worn off. Some have reported success with rubbing the
end with a pencil to coat it with graphite. A small circle of tinfoil
and a dab of glue might work too.