I think it'd definitely possible, though also somewhat unconventional. As someone whose father breeds canaries for song I can see complications with it, including that birds can actually go into shock and die from stress. A good way to combat this, of course, is get a hold of them from egg/very young and grow them relatively tame. That way they won't freak out whilst you are gardening and you could potentially create your own sort of ecosystem.
You could also create entry and exit points for small birds to enter through freely. Generally small birds do less damage to your plants, even if they aren't the birds you necessarily look for. And if you feed them on a schedule, they will learn when they can enter and when it's less fortious. You could train them by putting small feeders just inside of the holes (so they aren't actually inside) and teaching them to come to your place to eat.
Birds are very skittish creatures, but when they get used to you they can be really friendly and some species will even teach their young where it's safe for them to go. My aunt used to feed them on a schedule and the birds would actually peck the window or chirp really loudly if she was late. These were wild birds, mind you. She just loved seeing them around.
But either way I definitely could see this happening, though it'd be fairly experimental. I personally love the idea and with the right care it could be quite fun. I kind of imagine it to be like a butterfly house, except with edible plants and birds