@Chad Welcome to the forum community and thank you for your contribution to this topic. It’s a very interesting question!
We have housesat for horses via this site and outside of the site and have been lucky enough to ride on two occasions, but this was after the pet parents had seen videos of me riding previously and one was a repeat sit where I had a riding lesson with the riding school next door first so they saw how I rode. I’m lucky enough that I used to own and work with horses previously as an event groom, but we would never expect to ride and on both occasions went in a non-riding capacity and got offered by the owner to ride at a later date, once they knew us more.
We have done sits that are non-riding and either just fed and watered in the field or some have included mucking out, but as I am unable to own my own horse at the moment even mucking out with all of the smells etc takes me to my happy place!
We both work remotely and if there were too many horses to muck out say in winter it might not work for us, but normally the care requirements for horses are pretty clear on the listing and the owner will ask if they require someone with horse experience or not.
I’m not sure of any horse-centric sites that work like this one, if you find one let me know! Facebook groups can be good at finding similar horse opportunities, but they might not quite work on the same ethos as THS. I would say not to think you can ride with any horse housesits and if the owners offer that is a massive bonus!
This is a great topic on the subject and discusses things like insurance:
I hope you manage to find a suitable horse sit 