Another thing worth mentioning is the false analogy of "clipping feathers is just like clipping nails, they grow back".
I'm not a flight expert, but I got into birds because they're actual dinosaurs (coelurosaurs, group that includes T.rex and Velociraptor, excludes Allosaurus) and I've been passionate about dinosaurs since I was 3 years old. I am 29 now. So dinosaur evolution is an area of interest to me. And when looking at how many of the features evolved in conjunction with the emergence of flight feathers.... I gotta say that there's really no actual analog of flight feathers in mammals. Closest thing I can think of is the membrane of bats in terms of analogy.
Yes, they're dead keratinous integuments that regrow like hair or nails do. But flight feathers -- not down feathers -- have shaped the evolution of pennaraptoran dinosaurs for roughly 170 million years (we haven't found one this old yet, but we know they must have diverged then based on what dinosaurs are closely related, appeared around that time, but don't have flight feathers). Pennaraptora are all dinosaurs with pennaceous feathers (Velociraptor, Microraptor, modern birds, not T.rex). The reduction of their tail, lightening of their heads, increase in cognitive capacity and higher sensory functioning, digestion (frequency of pooping), their flight-or-fight response, wrist flexibility (folding wings -- this was an increasing trend we saw in theropod dinosaurs long before the first birds appeared, but bird took that ability to an extreme), digit reduction, etc. Have all evolved as a consequence of, or in conjunction with, their ability to learn to fly. Granted, flight isn't the reason neornithes (the branch of dinosaurs all modern birds belong to) were the only ones to survive the extinction event 66 million years ago. It may have actually had something to do with reproduction, because no other branch closely related to neornithes survived. But it's still an incredibly central part in terms of selection pressure in birds. Of course, some birds secondarily evolved flightlesness. But these weren't just clipped from one day to the next. This was a consequence of generations of adaption.
Their entire evolution depended so much on their ability to fly. They serve such a primary purpose of locomotion that you can't just expect to remove their ability to fly and think it couldn't massively impact their life in some way. A shaved dog may be disadvantaged too, but it can still move the way its entire body intends it to move. A parrot? No way.
And yeah, I'm trying to avoid making an appeal to nature here. My point isn't "bird flight is natural, therefore good". More, it's so fundamental to their entire functioning, that you can't just expect to deprive them of that without any secondary consequences. I think a much closer analogy to wing-clipping is a lobotomy in humans, in how our brains impact and define our functioning.
I'm not trying to show off what I know about dinosaurs or bird ancestors here. Just providing context to the importance in the emergence of flight and flight-feathers.