If you have the resources, there is no reason you can't get a horse solely to be a big ol lovebug pet.
A few months out of the year Indie has a break to recoup from the long trails during the summer and early autumn riding seasons... I gain a lot of joy and companionship out of just caring for her and taking her on the occasional hand-walk.
I think it would be very sweet and kind hearted of you to take in some of the unridable horses that few people have the resources to simply have sitting around. I think you'd find they can be just as much of a joy as any other pet! And if you're a horse lover, more so ;) Post pictures if you get some, I'd love to see the cuddle-bugs.