Hi Twisty,
My daughter doesn't mind doing it herself at all and in fact won't let us do it now. She did get into the habit of rubbing the area before she put the needle in - which is supposed to help, but now she doesn't bother as she wants to get it done quickly so she can get on and eat!
We have never used a pen mate, but I'm sure that any click noise must be more scary than the injection itself. My son has in fact just gone onto an insulin pump which has a firing mechanism to insert the cannula and he starts crying in anticipation as he is waiting for the 'bang' as he calls it. We always ask him after we've inserted it if it hurts to which he always replies no.
As for doing it herself then I think it's an individual thing. She has also not been diagnosed for very long so is still probably adapting to her new lifestyle. I wouldn't worry about it. I heard some children doing it as young as 3 and other children when they are about 11. When she's ready she will tell you.