A good phlebotomist is worth his or her weight in gold. Venipuncture is one of those things someone is either good at, or not. There's usually not an in-between.
My mother is a retired lab tech and phlebotomist and for many years, she was the only one I allowed to draw blood from me. She was good at it. She had the "touch," and it's something you have to have if you want to be good at your job and not hurt people, however unintentionally.
If you find that you're routinely hurting people as you're training, you probably need to find something else to do.