Thanks for the article. It was interesting and does point out a lot of problems but I don't think it makes a satisfying enough case for all horse riding being a problem.

Most of it hinges on two pieces: the first discusses alterations in the spinal processes (the narrow projections on top) of horses' back vertebrae. The article states,

In 2007, Matilda Homer and colleagues conducted a study with 295 horses who were considered physically sound before the examination. In this study, x-rays revealed that 91.5% of the horses were diagnosed with alteration on the spinal processes. Almost always, the spinal processes of the caudal saddle position were affected. The most frequent results were diminished internal spaces of spinal processes, including changes of the bone structure of the spinal processes [1].

With the implication that this constitutes spinal damage.

But that's not what the study concludes. From the abstract:

In spite of considerable x-ray results some horses show no clinical discomfort. On the other hand alterations on the spinal processes can cause pain and involve problems when riding the horse or make a horse even unridable. In order to being better equipped to assess the x-ray results on the spinal processes, a retrospective study was performed by way of analysis of spinal processes -x-rays of 295 clinically back sound warmblood horses.

[...]

X-ray alterations thus are frequent even among clinically back healthy horses. If alterations occur almost always the spinal processes of the caudal saddle position are affected. Recent court decisions have ruled x-ray alterations not accompanied by clinical symptoms as not being material defects. Therefore it is worth discussing whether an x-ray exam as part of a general exam at the time of a purchase makes sense at all under these circumstances when x-rays are performed without a clinical suspicion of a dorsal ailment it would most probably be sufficient to perform a singular x-ray of the caudal saddle position allowing for radiation protection.

While the spinal abnormalities, which to be clear are just deviations in shape from horses that aren't ridden, can cause pain, the authors do not assert that they always do or that they even constitute injuries.

The other piece discusses bone fusion over a horse's lifespan in the context of horse racing and concludes that the commonly held assumption that horses are safe to ride once they look like their skeleton is mature is inaccurate, and that they should be ridden only once their skeleton has matured further:

Just about everybody has heard of the horse’s “growth plates”, and commonly when I ask them, people tell me that the “growth plates” are somewhere around the horse’s knees (actually the ones people mean are located at the bottom of the radius-ulna bone just above the knee). This is what gives rise to the saying that, before riding the horse, it’s best to wait “until his knees close” (i.e., until the growth plates convert from cartilage to bone, fusing the epiphysis or bone-end to the diaphysis or bone-shaft). What people often don’t realize is that there is a “growth plate” on either end of every bone behind the skull, and in the case of some bones (like the pelvis or vertebrae, which have many “corners”) there are multiple growth plates.

So do you then have to wait until all these growth plates convert to bone? No. But the longer you wait, the safer you’ll be. Owners and trainers need to realize there’s an easy-to-remember general schedule of fusion – and then make their decision as to when to ride the horse based on that rather than on the external appearance of the horse. For there are some breeds of horse – the Quarter Horse is the premier among these – which have been bred in such a manner as to look mature long before they actually are mature. This puts these horses in jeopardy from people who are either ignorant of the closure schedule, or more interested in their own schedule (for futurities or other competition) than they are in the welfare of the animal.

Horse racing and other activities that pushes horses to their physical limits are clearly unethical and should be abolished. Bits, too, seem fucked up based on what the article described, but they don't cite any sources for that.

But all that aside, even if we assume that, yes, it's always cruel to ride a horse (which I still don't totally buy), horses were just one example. My main point was that the idea that it's inherently wrong to use an animal for any purpose is absurd. If you can't accept that riding a horse can be ok, then substitute dogs or llamas.

It's this line from the piece,

Horses do not exist so we can ride them. Just because we can, and because we desire to, does not mean we have the right to. We do not have the right to exploit any non-human animal for anything.

that doesn't make any sense to me. What does "exploit" mean here and what's inherently wrong about using an animal for a purpose?

The emphasis on us lacking some natural right to use animals for "anything" is, like, classic liberal reasoning.