This is an interesting question.

But I have not yet encountered Chariot Archers, only War Chariots... I think they did not fire arrows, they threw spears or something like that.

This is ridiculous

Were Chariots not there for mowing down Infantry? - Egyptian Chariots e.g.

In RTW, they butcher Cavalry, but no Infantry. While Cavalry butchers Spearmen?

I have my doubts about this kind of balance!