wow this sounded so much like me when I was younger I felt compelled to respond. I'm sorry you are having a difficult time but I think it's wonderful you are seeking advice. so many parents just ignore these types of issues, or just go along struggling and resisting it causes so much frustration for both the parents and the child.
when I was a child I was always very independent, I excelled quickly at things and had varied interests including music and art, but I had an extreme fear of failure. whenever things got difficult I wanted to quit, not because of the challenge really, but because I was deathly afraid of being embarrassed. I can see it now as very low self-esteem. my mother tried very hard and was very loving, but I think because of this she gave into all my whims. that meant trying new things and then quitting when I was scared.
Some of the things I quit were piano, violin, flute, softball, cheerleading. I can honestly say quitting was the worst thing I could have done. It created so much drama and everyone trying to convince me not to, it was always a drawn out thing with a lot of anxiety and I felt like I was letting people down.
I'm not suggesting parents force their children to do activities when they are really imposing their own needs and wants on the child, living vicariously through them or using it as a way to subconsciously compete with others. But there is something to be said for teaching your kids to follow through once they've committed to something. Not forever, but setting a realistic period of time is good. Once they've made it through the committment (a season, a certain number of lessons, a specified number of months) they've fulfilled their obligation and exposed to something different whether they succeeded at it or not.
I would really caution against letting your child quit something MID-committment. It's like not making them try a food they don't want without tasting it first.
If he is showing a real 'fear' be patient and encouraging without enabling them.
Find out what the real issue is. Sometimes at that age they still have a limited attention span and some personalities just always need to be challenged. Sometimes it is a real fear of failure (like I had and it sounds like you are aware of) that manifests itself at an early age and is disguised this way.
I believe the ONLY real way to deal with this is to allow them to fail, but they need to know you will be there to catch them when they fall. We want to always shelter our children from pain and embarrassment, we want them to like us. But more importantly, we want them to TRUST us. Failing is a part of life, it's not the failing that affects their self-esteem, it's how they deal with it and recover from it.
I am still dealing with perfectionism as an adult, and it used to be something that I would laugh off and roll my eyes at, but now I am realizing it is a real road block to life. It was an excuse to not do so much that I was scared to do. It is an unrealistic way to approach life and for anyone that is prone to depression or anxiety, it is an awful crutch.
I know this was long and winded, but I hope you are able to find the root of this and use it as a lesson in life for your son. I wish you luck!