hi, first i can see from your video you really care for this bird.i'm going to tell you what i did when i inherited my amazon who was in a similar situation.my uncle bought him then married a woman who hated the bird so he dumped him with my grandmother who had problems his diet was horrible,he was always caged.when she passed my aunt took her house and put him in a closet when my mother visited months after my grandmother died she brought him to our house he was gray and covered in brown dander and even cobwebs.we imediatly gave him a bath and did our best to care for him then my mother screwed up and let my alcoholic brother the bird he had him for a few months before getting into a fight with his girlfriend and coming home to mom.he went home and left cebal, i heard my mother complaining about cleaning his cage so i said put him in my room.he was completely cage bound would bite if i took him out so i lured him out with food and closed the cage and gave him total freedom he even slept on a nightstand next to my bed.i tried to get him to step up but everytime he bit so i would just sit with my hand out palm up until he got curious and explored my hand still i did not move to touch him for a while then tested him once in a while and finally he did come up after we built trust everything came easy i never forced anything on him.as for your budgies i would move the cage closer so they can get to know eachother without contact.if possible let them out in another room for a while.when you do let them out in the same room keep the smaller birds away from the gray shoo them away i wouldn,t shoo the larger bird to prevent the larger gray from resenting the smaller which is how i got mya

used to penny

now she doesn't lunge at him when he happens to land next to her.sorry it's so long but i thought it might help to know the whole history so you can see there is hope for your gray.