Perhaps that a recorded version of the song is the same each time you listen to it, whereas listening to multiple live performances of the same song will capture the improvisation/variance that is so important.
If that's true, then much of jazz music should have never translated on recordings. It also was not uncommon for multiple takes of the same song be recorded for jazz as well. I feel like that should apply here, unless there's another distinguishing factor.
Jazz (which I am more familiar with) is a reasonable analogy. And I agree. You should be able to capture a good studio performance just the way Ornette Coleman and Coltrane could.
One thing to note is that jazz has more structure. There is improvisation around that structure. Indian classical music is structured around rhythms and scales (and Carnatic music is microtonal) so it gets pretty out there.
recorded live performances were mentioned, but I could see how hearing multiple recordings of a particular song in a style that is known to change performance to performance could give you more appreciation, I'm just not sure how multiple recorded live performances differ from multiple studio recordings. Maybe the implication was that there are multiple recorded live performances for most popular artists compared to a single or very few studio recordings from that artist for a song, and so it's worth listening to the good live recordings to hear the variation?