This sounds familiar. We ended up with a wild pigeon once. It had been injured and had a broken wing. The wildlife rehabilitation facility we consulted told us she’d never be able to be released, told us how to care for the wing, and what to feed. The first six months or so, she fluttered all over her cage when we would fill food/water and clean. After that, she calmed down and was fine with us near her. After about a year, she started hopping up and cooing when we’d come near her cage. She eventually became completely tame. Eight years after rescue, she laid a clutch of eggs - shocked us! (That’s how we determined she was female) She regularly would sit on my daughter’s desk next to her computer when the daughter was working. She loved to perch on my daughter’s arm and be carried around. She definitely had a happy life.