I remember too! I am in my sixties, and I grew up in Michigan. I remember that song, and can still sing the refrain--and you've got the words mostly correct. I think. Except that I believe the exact words were:

"A nation of immigrants, a nation of immigrants
Built this land of opportunity
Descendents of immigrants, of hard-working immigrants
Are we"

I came across your comment just now because I myself have been trying to track down that song, and information about the program(s) it was from.

I remember a TV series which I believe was called "Exploring" from that time. I believe it was broadcast on Saturday afternoons, and I remember that one documentary-type episode was all about Ellis Island. It's my impression that the song was used in that episode, and was perhaps the theme song for the series from then on. Well, I could be wrong about that, but I DO remember being fascinated by the program about Ellis island, and I DO remember that song.

Like you, I've had absolutely no luck in locating the song.I've Googled everything I can think of. I'm very happy to know that someone else remembers it!