Hi and welcome to the forum
I’d leave him to choose when to come out.
The idea behind a hand tame budgie is that you have a bit of a head start in taming as the bird is not fearful of hands. He isn’t bonded to you and he is still scared, (just not particularly of those grabby hand things - He’s had a more balanced introduction to those things) his whole world has changed.
When you brought home your bird, he did not know his cage as his safe place. You were the closest thing he recognised and he submit. Now he’s accepting the cage as his home in this scary new world.
Leave him to settle in to his new environment. Move gently when you need to be inside the cage to change dishes and clean. Hang out around the cage, chat to him & offer him treats.
When he’s consistently happy moving around his new home, chirping and preening beside you and coming straight over to you when he notices you holding a treat, then you can start tempting him outside. You could then consider moving his food bowls lower down, to get him used to being near the door. Followed by placing a perch on the outside of the doorway, and attaching/holding a treat there. Leave the door open, sit back and let him take the lead.
It’s about trust not dominance. He isn’t being naughty, he’s scared. You know that you are safe, he doesn’t know that.
There are many different approaches to taming but my line of thinking is - Do we really want to bully our birds into compliance?
The bird has 3 responses to fear - submission, flight and fight.
Some people (your breeder?) would suggest forcing them out with the logic, “Then they’ll see it’s ok!” So, the bird then flies off in a panic.
The owner is then told, “Its easier to tame them with their wings clipped!”
You’ve removed flight, so now the bird is left with the options of submit or bite you.
So then we get posts, “Help. I was sold a supposedly hand tame bird but now it’s biting me!”
This isn’t the best start to a good trusting relationship.
They’re not like dogs, birds are much more wild. It’s a completely different relationship
Ultimately the question is: Do you want a bird who is forced to interact with you or one who chooses to?
My advice is to read through our new bird arrivals and bonding and taming stickies. You DO have a benefit with a hand tame bird, but all birds are individuals and adjust at different rates.
I’ve just reread this post and realised that it sounds much more abrupt than I intended. I apologise for that, I think that half way through, I inadvertently slipped onto a soapbox and my reply came out more of a rant than a response. My irritation was with the breeder’s approach and not at all with you.