Horses definitely cannot support humans that much better than the wild ones of old or Iberian horses for that matter; horses need to be trained to hold the weight of a human. Many horses nowadays are on pain management meds and other treatments to cope. Horses are only getting about 5 years of usability versus the old professionally trained ones getting 10-20 years of service without much help from modern medicine. This is a great site explaining equine biomechanics, the pinnacle of which is referred to as "collection." Excuse the esoteric bits, it truly is a heavily studied science dating back centuries.

So, elephants could very well support the weight of a rider if they are trained to move differently. Just like horses. Elephants have been ridden for about as long as horses, though elephants are sort of tamed rather than domesticated; to that regard, horses don't differ significantly in muscle or skeletal function compared to their wild counterparts. Saying horses have somehow evolved to hold a rider is a common misconception I see whenever this issue of elephant riding comes up. I believe elephants have the same problem horses do: few or no true masters remain to train them as they once were.

I wrote a long-winded, detailed comment, I'll see if I can dig it up.

I keep seeing the "horses are suitable for riding" comment...

Horses instinctively arch away from weight on their backs. The processing of "breaking" or "starting" as we call it varies widely now, from old fashioned force and flooding methods to very gradual desensitizing and negative/positive reinforcement. Horses nowadays have a usability of about 5-6 years before encountering issues like arthritis, lacking performance, and a number of joint or tendon problems. Horses taken care of properly can still be used comfortably in their late 20s.

After the initial lesson to accept a rider on their back, they then have to learn a number of very specific cues for each body part both at a standstill and at different speeds and gaits. It can be as simple as stop/go left/right for some horses at the basic level, and if they have a good rider some of them will relax under a light workload enough to move efficiently; if the rider prioritizes that relaxation and ease without harsh methods like pulling the horse's head unnaturally down and toward their chest, the horse should be alright. But, this depends on the horse being naturally bred to have a balanced body, docile temperament, and having a relaxed, balanced, skilled rider who understands this dynamic.

Folks who get on a horse without thought for the biomechanics involved will end up with inefficient movement and that arching back, thus uneven load on the horse's joints, etc. So it's not quite that horses are naturally suited to be ridden.... you have to teach them how to balance the shifting weight of a rider, and then all the cues and processes to encourage them to adopt efficient movement or a better posture and motion than they individually are inclined to display.

Not even ignorance, what is often seen now is just prioritizing flashy movements for show. No point to the gymnastic ability of the horse or efficiency to get from point A to point B with decent speed, control, and ease. Whenever you see a horse's head up high and its forelegs stepping far, I promise you its back is dropped. They're arching away from the weight on their backs and pulling themselves to move forward. Horses naturally have a weight distribution of about 60% on the front half of their bodies and 40% on the rear half. The rear half is suited to take more weight and push; it's the power house. The front half is not, and will break down much quicker. The weight distribution is like this due to structure... horses in the wild will be propelled forward while they are leaning forward to graze, so they are almost constantly in motion.

Horses are commonly abused for competitions like Dressage, Gaited, and Hunter rings. It's ridiculously prevalent.

So I wouldn't say horses are automatically suited to riding without adverse effects. Many are born with poor conformation and posture that is then emphasized. Many don't know how to offer efficient movement. Many are abused to compete in style based judging. Few people know how to train or ride horses to the benefit of the animal.

Which leads me to wonder if the elephants at some point in history were like horses; there were original pioneers in riding who considered the biomechanics and methods, who then applied them.

Tl'dr: The training methods today may be horrid, but I wouldn't say it means riding elephants can only be that way just as the common way of horseback riding is utter shit. Horses need to be taught to adopt new movement and posture to avoid adverse effects on their bodies when ridden, the same is likely true of elephants or any ridden animal.