You are being taught to protect yourself from injury -- and pain thresholds are one of the ways your body tells you that you are being injured, and it could get worse so do something about it. (A horrible snapping noise is another).
Tapping out is a good thing to have in your set of skills -- I personally recommend learning to tap with every limb...I've been known to flop like a fish on occasion
(I think it started when I made the mistake of rolling over and trapping my free hand under my body -- not a bright move, by the way) .
All of the joint locks are painful -- it is the pain, or threat of pain, that prevents you from getting up and attacking again. Some of the pins are functional, rather than painful -- you literally cannot move, although you are not currently in pain. The Nikkyu and Ikkyu style pins can be like this. Also, you can practice aikido without using joint locks -- performing kokyuho or kokyunage techniques, and throwing your partner to the mat rather than trying to control them and pin them.
As for Nikkyo (the lock, not the pin) not being painful when applied by the "masters," IMHO, some one has been feeding you bullpucky. You can apply Nikkyo in a way that it is not immediately painful -- but then you are not applying any pressure on the lock, and it serves no purpose -- you may as well perform ikkyo.
I have felt Nikkyo put on my wrist in such a way that my resistance to it was what caused the pain -- and it's pain and the threat of more pain that motivates uke not to attack further. But it still hurts.