You are doing wonderfully with Tui.
One thing you cannot do with a grey or really any parrot or animal, is force them to like something they are fearful of. It is only accomplished in itsy bitsy desensitization over a long period of time as you may progress one step or inch closer each day week and just stand there praising them for being so bold. The little baby happy to see mommy may never go away, just ignore it or perhaps offer a foot toy you keep at hand to divert that attention. My wife has to ignore Dayo sometimes for 30 seconds or so because he is so happy to see her he will almost instantaneously start the gurging reflex.
Right now, the way I see it based on your description, is Tui has two "Safe Zones", his cage and you. That's as much as he can trust anything right now. Thankfully he does trust you. So really, the only way to keep him feeling happy and safe is just spend the time you can with him, then place him back in his cage if you do not want him to come and bug you again until you have the time for later.