A different life doesn't necessarily mean a bad life.
For some birds, it certainly can, for whatever reason their needs cannot be met and they are unhappy. They don't adapt, and despite the fact their lives would be much shorter if they lived in the wild they likely wouldn't be unhappy.
Many birds do adapt and are happy though. Would that same bird be more happy if they had been born and grew up in the wild, certain instincts more fulfilled? Less happy because they don't have all those fun human inventions to break, food they would never encounter?
I dunno. I can't know.
The real trouble comes in when I see neglected birds or birds who are obviously unhappy. What do you do with that unhappy bird who cannot and does not want to adapt?
Why do humans bring an animal into our homes and than treat them like garbage? If they aren't even going to try and meet their most basic needs, yes, that bird would almost certainly be happier and more fulfilled in the wild. It may die to a hawk after raising one clutch of chicks but that year or two it would be alive would probably be miles better than being kept in someone's basement; alone in the dark and with nothing to do.
While I don't think you can compare a wild animal exactly to a domestic one or a human, I do generally think of it like this;
Some people will be miserable living in a big city, working 9-5 5 days a week. They would prefer a simpler life, maybe with fewer amenities many of us take for granted, but living rurally and working for themselves.
This is an attitude you see a lot in "the bush," rural Alaska. People would rather live off the grid, hunting for their food and bartering for everything else, if they have a job it is usually an extremely local job. It is not easy and there is little health care, any serious medical emergency usually means a helicopter ride to Anchorage or Fairbanks.
Yet I meet many people who grew up in the bush and loath that lifestyle. As soon as they could leave home they moved to the city, many want to leave the state for good once they graduate college. They want the stable job and big city. TV, videogames, groceries, toilets that are not a bucket.
Neither one is wrong, just some people will be happier in certain situations. They have their own individual likes, dislikes, needs and wants.
Parrots do too. The difference is, unlike the human example, one situation is unnatural, one is natural. But like I said, different doesn't have to mean bad.
I'm sorry you feel depressed at times. If you ever feel yourself getting too down, try to think about the things they love and enjoy despite being in your home instead of the wild. Be grateful they are in your home with a human who really cares about their happiness and tries to give them the best life they can have.