I think cutting or keeping is not the actually point to decide over, here.
Like many suggested, simply start a serious discussion about how to improve and diverse the mounted line. I think a lot of good ideas have already been mentioned here and a clean, guided discussion about how to improve might be the best thing to do now.
I think seZereth is completely correct here, that this is more about how to handle mounted units than about chariots specifically.
Personally, I liked the idea that has been proposed of splitting the mounted line into light (faster, greater withdrawal, weaker attack) and heavy (higher attack, slower, less chance for withdrawal). The base mounted unit could remain the horseman (who should take weapons promotions) and then allow for the split, like TheJopa suggested.
The light line should be about scounting, harrasing weak enemies, cutting the lines of supply. It could be given access to sight promotions, given flanking damage powers against siege and archers, increased pillage yields, and at high levels, possibly the marksman promotion. It could have its own NU.
The heavy line should be about battle in the field. High attack, decent defense in the field, but penalties for defending in cities and rough terrain. Attack bonuses against melee units and seige weapons. At higher levels, it should have access to blitz. In fact, having blitz could be the big bonus for the upper end heavy mounted units, given how rare blitz is in this mod.
Also, given the lore, perhaps the Hippus only could be given access to a weak charriot very early on and without building restrictions, at exploration. Perhaps a 3/2/3 unit, giving them a use for their palace provided horses long before they get to horse ridding.
The charriot art could be kept, specially for the dwarves, given that lorewise they are master of craft, and not likely to get on horses. Charriots and warcharriots could be their heavy line, while the boar riders remain their light mounted line.