From Chicago blues man McKinley Mitchell, The End of the Rainbow.
Rainbow Stew - Merle Haggard
Ween sang about the Homo Rainbow.
Make Me Some Rainbows - Ella Fitzgerald
Silver, Blue and Gold by Bad Company sings about the fact that his rainbow is overdue.
Robbie Robertson- Sign of the Rainbow
Good rainbow memory. I love that Bad Company song. They were my first concert. They were the headliners with two other groups.
Seems to be many more songs about rainbows than about unicorns. And definitely more than songs featuring humpty-backed camels.
https://www.youtube.com/watch?v=qRC4Vk6kisY Pink Fluffy Unicorns Dancing on Rainbows
Rainbow In Your Eyes - Al Jarreau
BTW, when looking for a video for the OP, I discovered that Kenny Loggins did a cover of Rainbow Connection. Words don’t exist to express how bad it is.
Danny Hutton’s Roses and Rainbows from 1965. One of my favorites as a kid.
The Rainbow - Gene Autry
I’m surprised this has not made the list yet:
Israel Kamakawiwo’ole 
‘Over The Rainbow’ & ‘What A Wonderful World’ Medley 1993
The OP’s question has been very well answered, but I can’t help but think all songs post-1979 should not be counted for the purpose of this thread. Kermit/Paul Williams was referring to songs about rainbows that already existed. 
“Love is Blue” (1967) contains the phrase, “When we met, the bright sun shone/Then love died, now the rainbow is gone.”
With that cover art, words don’t need to say a thing.
Just so you two cynics know, the album Return To Pooh Corner was described (in the link) as, “…music for parents and children to enjoy together.”
There are some very famous contributors to the album. Also from the link, “It was a successful album for Loggins, selling over 500,000 copies, and was nominated for a Grammy Award for Best Musical Album for Children.”
Yeah, I thought about that myself, after the bell was rung.