I'm not surprised you have trouble approaching a random girl. It would have to be the most difficult way of meeting a woman and quite likely to fail. It makes it even more difficult as you're only interested in the pretty ones - they get hit on all the time and by men who have more confidence than you. Because of this, pretty girls would also have far more experience in saying 'no'.
As you say that you have confidence in all other areas, I'd suggest you stick to approaching girls who you already partially know. Friends of friends, girls you meet at sports clubs and events. Girls who frequent the same pub as you. Girls at college. In short, approach women who you have been exchanging smiles with when you see them.
I do also believe that your interest in appearance is holding you back. You talk about how people are shocked that you've never had a girlfriend and go on to talk about your looks. If we're talking about the initial attraction to appearance, I'd have to say that I was always attracted to an open and honest smile and nice eyes. Basically, I look for what type of personality shows through his face. Yeah, there's the odd hot guy I would have like to have shagged.....but good looks mean little in terms of a relationship and I think most girls know this. Even my 15yo announced yesterday that she's grown out of only being attracted to cute boys LOL
As for restricting yourself to good looking girls - what would you do if this good looking girl suffers a facial injury, gets sick and starts looking haggard or heaven forbid, gets old? If having eye candy is so important, you're going to find yourself struggling as you you age. Good looks are temporary - only a rare few really stay good looking for life. I know I'm being harsh, but you are restricting yourself so much.
Anyway, here's my dating strategy: Learn to smile at women you see frequently. Not just the pretty ones. If a girls starts to smile back when she sees you, introduce yourself and exchange names so that you can greet each other with words. Then if you find that she seems interested, then ask her out. Meeting her in a step by step manner is not only less scary, but more likely to succeed.
Never regret anything that has happened in your life. It cannot be changed, forgotten or undone. So, take it as a lesson learned and move on.