To me, it sounds like he doesn't know "how" to step up.
This may sound silly, but the floor is not the same as the perch. A bird may step up from *any* location within the house, but 'refuse' to step up from the cage. In the bird's mind, the cage is not the same thing as the house. Birds may also have a fear of hands near cages... hence the phrase "he's tame once you remove him from the cage". It should never be like this! NEVER!
Basically, just because the bird knows how to step up from one location (i.e. the floor) doesn't mean that the bird knows how to step up from another location (i.e. a perch or the cage).
He's not stubborn. It's just quite possible that he doesn't know what it is that you are asking of him! And the easiest way to "fix" that is to generalize the behavior. He steps up from the floor? GOOD! Treat! He steps up from a chair? GOOD! Treat! He steps up from the couch? GOOD! Treat!
Any place he will step up from, reward him for the behavior. Make sure he steps up in other areas as well as from different items. Teach him that stepping up is good and reward the behavior. This will thus make it easier to pick him up and move him.
You may also want to try training step up using Barbara Heidenreich's method.