A few thoughts having read through the thread.
1) I agree about playing through a mistake, although that is not the same as stopping and working on the troublesome part to prevent the mistake. What I have found is that when I know a piece really well, the mistakes don't matter as much, inasmuch as they don't stick out as bad because I am able to recover and keep going without stopping. Kind of like the difference between stumbling while running and using agility to keep going as opposed to falling down or stopping to prevent falling.
2) To get to that point, knowing the piece so well on both the conscious (cerebral cortex) and sub-conscious (cerebellum) level is key. When I pick up my guitar having not played it in a very long time, I am sometimes amazed that my fingers are going to places without me thinking about it, almost like I am watching myself play, because the cerebellum has the program stored. Problem is, once I stumble, I lose my way almost immediately, and once I begin the consciously think about it, I literally cannot play a section that I only played 30 seconds earlier, and have to look at the music and start to consciously remember the notes. In order to perform, I think you must have seeped in the piece at both levels; the conscious where you know every note, every measure in detail, as well as have the cerebellar program to carry you through and be heading places with your fingers before you are even consciously aware. This allows you to continue even when distracted by the thoughts of "all those eyes".
3) Sort of to that point, I remember seeing Michael Lorimer perform when I was in college in the late 70s. He was very engaging and spoke to the audience about the piece, the Baroque guitar he was playing, etc. But when he began playing, he went to a different place...to the point where when he finished, it was literally like he was coming out of a trance...it took him a full second or 2 to sort of wake up or return. That really made an impression on me.
I have pretty terrible performance anxiety, but I believe I can learn to get past it. It's a lot like public speaking; the better you know the source material, the less concerned you are about making mistakes because you really KNOW what you are talking about, as opposed to merely having memorized it. Memorizing a piece is not the same as knowing it (point 2 above) - just the same as memorizing the Italian-English dictionary will not allow you to speak Italian.