It comes down to one main thing, but there are other factors that probably played a part too. The main thing is that despite the nature of the Marshall attack, it is considered drawish at the highest levels. At lower levels, of course, it is a double-edged game where anything can happen. And while they are double-edged, white is on the defensive, not where Kasparov liked to be.
There is, clearly, a lot of theory to learn, but it is doubtful that Kasparov was put off by that since he had the reputation as the most prepared player in history at that time.
With the advent of computers, the Marshal has been analyzed to the N-th degree (meaning a lot), and most of the time, black has enough initiative to hold the draw, or better if white is not prepared. Kasparov, being Kasparov, probably felt like "why should I allow that possibility when I can probably play a line that you do not get the initiative, and we keep more pieces on the board, and I can simply outplay you in the long run? I am Kasparov, after all."
Lastly, and I touched on this already, but it is simply not Kasparov's style: He is the player, who loves to play with the initiative, so why play into a line that neutralizes his greatest strength? He did not, and I confirmed that by searching Mega 2020, and you are correct: There is not one game in there where he allowed it, and that includes faster time controls. That is not a practical way to play good chess.