Yes, most of my flock are 2nd (or 3rd or 4th or even 5th!) hand birds. All 5 of my lovebirds fit into that category, 1 of them my daughter has tamed and kept for her special pet, the rest we have in a large flight cage where all their needs are met (and then some!) and they'll live out their lives happily here.
My CAG is also a 2nd-hand bird, we got him (he's really a hen) a year ago when he was 4. His former owner obviously doted on him because he's very cuddly and sweet and gentle. In fact he insists upon LOTS of cuddle time.
My Meyer's parrot is now 4 years old. I'm his 5th home, and he was a year old when I bought him. He's my cuddliest bird. Within 2 days he had bonded strongly to me, and is still my most special little pal out of all the birds. He's totally sweet and wonderful with me, and will do anything for me...although he's definitely a one-person bird.
Both of my cockatiels are rehomed birds. Spiky - the hen - is totally social and lovable and gentle. Pokey came to us at the age of 8, strongly bonded to his mirrors and completely cagebound. He's now 22 and has mellowed somewhat but is still not a "people bird" although I do bring him out of his cage to spend time on the floor of the playstand every day.
I would have to say that from my perspective rehomed birds can be really rewarding and - forgive the anthromorphosism - appreciative of what they have in a new home.