The “Riding” skill could cover more than just mounted combat. It could be linked to actually controlling the animal or handling it in very dangerous situations.
I imagine that the poor Horse would be very hesitant to let something clearly unnatural as a Netherworld entity approach it; you might include a factor where the Horse actually tries to throw the rider off in fear depending on combative interactions with certain entities. The higher your Riding skill, the better control you have over the animal.
Certain mutations, especially the predatory ones might have negative modifiers when interacting with the animal as well, reducing your skill in riding dramatically.
But it can have positives as well; better negotiation over difficult terrain, knowing how to push your Horse faster to make the best use of its Stamina meter for Galloping.
Have the skill linked to crafting recipes for Tailoring barding/armour for your steed, or saddlebags to carry extra gear.
Better damage when trampling small enemies as you Gallop through them.
And later, an ability to call your Horse over to you with a Whistle if it’s within earshot?
Just a few hairbrained ideas that came to mind over the whole thing.