I adopted a 3 year old red throated conure who was terrified of being handled, but so desperately desired attention. She didn't require vet care, since she was seen by an a-vet and cleared as healthy before coming to me, but still.
If I walked over to her, she would run as far away from me as possible. When I walked away from her, she'd try to follow me without leaving her cage. Since I had very little reason to physically remove her from her cage, I let her be! Instead, I moved the cage around the house with me so that she could be near me, but still have the safety and comfort of her cage. She often enjoyed traveling down the side of the cage to get closer to me, but still didn't want to be touched. At that point, I put a perch on the outside of the cage so she could sit near me but again, not be on me.
After a couple of weeks of doing this, she finally got brave enough to climb onto my shoulder. Once we got to that point, it was a matter of staying near her cage because whenever she felt uncomfortable, she wanted back to her cage, and I would take her back! At first, she wanted to spend a few minutes at her cage to settle down before coming back to me. Then it was enough to touch her cage, stand on it, then climb back to me! Now? Instead of going to her cage to seek comfort, she comes to me!
I also allowed her wings to grow out, so she has the choice to be with me or be on her cage, and she prefers being on me!