I am a bit confused. Are we talking about the Christine McVie video album? If so, Songbird is played at the end. She tells everyone Merry Christmas and will see everyone on the road soon.
Maybe other copies don't have this song? I have the VHS tape and it includes songbird.
Is the injured finger speculation? Does anyone know if anything happened and how?
I think they just wanted to shorten the set a bit and decided to drop one encore and another song.
Songbird solo Country Club December 1983
https://www.youtube.com/watch?v=t9wkh82ceAk