I think pain is a varying element - not a constant. When I have new students they start out by feeling what they very often think is pain. It is just their body gearing up to meet the challenges of training. This will pass with time.
Next level is the locks and falls. When our body is not able to give a good attack and haven't learnt to relax during ukemi, then we feel pain from the locks that become a little too hard and the falls that we are not ready for. This will pass with time also.
There is of course the pain that we get from the odd mistake - an atemi that get's too close or a fall due to loosing balance in the wrong way. This - and the fact that some times you just meet stupid people that don't care if they hurt their uke - will become less frequent as we learn more and more.
The pain from knees and other parts of our body loosing their flexibility and ability to stand a two hours keiko.. well this will become more frequent as we grow older. But during the same time we learn to listen to our body and understand when pain is just at sign of our bodies being used, a warning signal or a body part crying out for help and relief... And we hopefully listen and comply to our situaion.
In some budo's / MA's you learn to ignore pain. In Aikido - the whole mental pain issue aside - I think we become pain-afficinados. We learn to understand a larger variety of pain and assess the 'quality' of a bodily sensaition, and we learn to deal with it, expect it and even welcome it as we link thousands of yonkyos (and other techniques) with the good feeling of training and the rush of endorphines that our joyfull activity gives us.
We are a rare bread indeed. Pain is only neccesary to the least possible extend, but getting to know the way your body experiences these inputs that will feel like pain for the first (many) years is a core concept of learning Aikido.
As one of my students say: Pain is mandatory, suffering is optional
It's not completely true, but it's not completely wrong either.