I think the same thing is true in ballet: it's the big jumps, multiple turns, especially fouettes and those that change position and speed, hops on pointe, and big balances that get people clapping in the middle of the music. In ballet competitions, it's the big extensions as well that gets the teen crowd shrieking.
I think it's very telling that, in the Turischeva clip, despite its many examples of extended dancing, it is the more sport-oriented tumbling runs that gets the big response from the crowds.
The shift from piano to recorded instrumental happened sometime between the 1976 and 1980 Olympics. Here are two examples of the same gymnast, the great Nellie Kim, in those two Olympics, the first with piano and the second with instrumental:
Aside from the stylistic change in the type of movement -- a lot of the 1980 vocabulary would not have fit piano arrangements -- I think there's a lot more continuous movement that reflected the rhythm of the music in the 1976 clip with the piano. (She was also at her peak in this event in 1976, I think scoring a perfect 10 to win individual gold.) I'm reminded of how Balanchine would ask the pianist, especially in his special Monday class, to play chords, not music, because if the pianist was working, then the dancers would think they were, too.