I have a place on Teacher Training (Teach First) for when I graduate.
I used to be a very shy and quiet child, I had low confidence naturally. It is only since uni I have addressed it and to my uni friends I am pretty confident and they are shocked if I say I used to be quiet. I still have the good things about being quiet in me, like you say, being a good listener, patient and caring. There are days I like my own company and really in your face loud people irritate me!
To improve my quietness, I did loads and loads of things with children and work in a school, I got really confident that it is something I am good at and love. That, combined with being a lot better at making conversation and not worrying what people think of me, gave me the confidence and I really have changed a lot. Ifeel much better about myself from diving into opportunities and meeting new people and not being the quiet one in the corner.
I knew teaching was what I wanted to do and that I am capable of it- but that to get on a course I had to also SHOW the assessors this.
Anyway, going off topic a bit!
In terms of being quiet and teaching, it is ok to be a quiet person, you have- as you point out- many attributes others lack! I had some quiet teachers and even though I'm more confident now, I'm never going to be a super loud, crazy teacher. But I will be an excellent one in my own style. You need to be confident in front of a class and be able to almost 'get in role' as a teacher - fake it until you make it!
Get some experience in schools to see if you can definitely see yourself doing what teachers do. Theres no shame in admitting in reality it isnt for you, there are lots of other jobs with kids. Are you confident but quiet? You need to be confident teaching to believe in what you are doing and be assertive.
You will be taught behaviour management, don't imagine it to be shouting at children- that is awful management! It is actually much subtler and quieter things like how you keep them engaged, wait for silence before talking, setting clear expectations, high standards from day 1 etc.
Sounds like you need to look into it more as to if it is for you- don't let being quiet put you off as long as you can see yourself doing it successfully and being confident.