I thought his primary complaint at the time was that they were reprogramming in the evenings in response to the day's games, providing a lot of grandmaster human input during the tournament. I could be wrong there.

I watched his later matches against Deep Junior, around 2004 (?) in New York City. Match was tied, in the final game Junior made a mid-game move that was surprising to everyone in the analysis room. They were using a different software to analyze the potential lines and not finding the advantage for DJ. Yasser Seirawan and Maurice Ashley couldn't 100% agree that it was a bad move, but they said from what they can see it looked like a mistake by Deep Junior. Kasparov to a lot of time to ponder, and they accepted an exchange that would lead to a draw.

It was a very psychological moment in that era when machines were not clearly superior to the best humans.

To some extent yes, but humans tend to make moves that follow some kind of reasoning or logic. When you play a strong opponent, it's possible that you won't see the next move that they play, but once they have played it, you can deduce the logic they used in order to make it. An engine move on the other hand can easily reject the standard strategies and can appear highly irrational. When you see this kind of move, it becomes easier to suspect cheating.

yes but the computer is just so much more powerful, as well as it makes seemingly weird moves more often, as well as it can get itself out of a "bad" line if pushed that way much easier.