I would prefer if you could raise your horse stats except impregnation by training them somehow (possibly by just riding them long enough but that's kinda vanilla) and instead their inherited quality determine how fast those stats raises.
The reason being that I will inetavitably bond with the first bunch of pixel in horse shape i tame even at base 10 and probably keep riding it until a bear maul us both to death, and so that interaction should be recognized a bit more, just saying