One year, a couple of wild house finches made nests in the hanging asparagus ferns I had on my front porch. Two females, each with her own potted fern, laid clutches of eggs in the ferns. I like birds, so I roped off the porch and put up a sign telling everyone to come to the back door until further notice, and called the birds "Lucy and Ethel" because of their idiotic twittering.
I didn't bother them until Lucy and Ethel and their chicks all flew away, then the porch was mine again. I wish they'd come back the next year, but they didn't.
Also nice of you. I find shelters to have a lot of loopholes when it comes to their no-kill rule, but sanctuaries meant specifically for birds seem to be a lot more caring, but either way, you're giving an injured animal to someone better equipped to help it and that's all you can do.