I have had similar thoughts to this concept of song performed better live than the studio version. How about songs that never had a studio version, some of the songs listed here may fit that category. The one song I have always thought of is
Jackson Browne - The Load Out/Stay (1977)
https://www.youtube.com/watch?v=rukvfk9a6rY
Stay is actually an older song that he paired with The Load Out.
“Stay” is a doo-wop song written by Maurice Williams and first recorded in 1960 by Williams with his group the Zodiacs.
The original recording of “Stay” was the shortest single ever to reach the top of the American record charts at that time, at 1 minute 36 seconds. (According to the record label, the Four Seasons’ version (see below) was even shorter, at 1:30, and remains the shortest charting record of all time, although it did not reach #1.)
There was a another song on juke boxes in the 80s that was about 30 seconds long. I can’t remember the name.
The live songs on Running on Empty are fantastic. It’s my favorite quasi-live album of all time. Criminally underated.
U2’s live version of “New Years Day” from Live Under a Blood Red Sky is far superior to the studio version on War.