The choke only opens all the way when you hit the throttle.
This is definitely not right. I assume an electric choke?
When it's warm the choke should always be open. Not just when you hit the throttle.
Sounds like choke needs adjustment or the bimetal spring in it is toast and needs replacement.
Make sure its cold and take the air cleaner off. The choke should be open. When you set the choke by pumping the gas, it should close so its only open about 1/8". Just turn the key on but don't start it. An electric choke should then open by itself anywhere between 30 and 60 seconds.
If its not opening with the key on after being set, you need to figure out why.
Check that its getting power with the key on. Also check the adjustment. If its not getting that 1/8" crack when being set it needs adjustment.
Could also have been put back together wrong after the rebuild.
Many times the black choke adjustment cover is not put back together so that it engages with the hook on the end of the bimetal spring after a rebuild.
Check out these things and then report back.