@REmasher centaurs, i mean having your stuff on full display is just as embarrassing as you make it out to be, if you don't think it's a big deal it isn't, they might have some barding while visiting major towns for the other races sake.
i mean they are basically refugee's so they just took whatever they had on hand.
TBH i just think neither the artist or writer actually thought about that very much overall to be honest, but the above is my mental explanation, so it's the one I'm running with.
and i imagine they use the horse parts in general, since it would be sucky for a centaur to have it on the human half.