The thing is he "doesn't" want to be picked up. He feels content in his cage, and he won't let you bring it around. Mine was very similar, though he would stay with me on my shoulder when i got around, he bited pretty hard when we wanted to give him a new batch of seeds in the morning. We went around this click training him: when he heard the click food was on the go and he eventually stopped biting.
My advice would be to click train or to pole train (where you train your bird to follow a colored stick) him. It can lead to him voluntarily step on your finger, you just have to have something to bribe him with. My tiel just wanted dry breadsticks, yours could prefer sunflower seeds or fruits?
Even though it's not nice, try to leave him without food for one night/day (not too long, we don't want to starve the poor thing) and then in the morning when he's hungry, offer the training. Just 10m per day, 2-3 times a day. Everytime offer him food, and it will lead to something. An important thing is to do the training FAR from his cage, or he will wanna go home and nothing else, and get distracted.
Place yourself in a nice calm room and arm yourself with patience. Lots of it.