Laurie Lewis
has an album of bird songs dedicated to the Audubon Canyon Ranch..a fundraiser for it, which I think has netted $10,000 or more
https://www.laurielewis.com/s_m/store/bird.html
She also sings a song When the Night Bird Sings...beautiful

Sally Van Meter (dobro player)sings a song called
(You Were) The Bird That I Held in My Hand ('Til I learned to Fly on My Own)
I think it is by T. Bone Burnett, of Oh, Brother fame.

I have done my radio show featuring songs about birds
and had no trouble coming up with 2 full hours.

Judy Fjell has a song Birds Return... I think it is the title song.

and here is a site with actual bird's songs---slowed down

http://www.hawkowlsnest.com/2007/11/slowed-down-bird-songs.html

Nanci Griffith has a beautiful love song -- Gulf Coast Highway

"And when she dies, she says, she'll catch some blackbird's wing. Then she will fly away to Heaven"

I am sure there are more...I could post the entire setlist from my radio show if you want....