Horse riding should lead to muscular backs but only if the rider's posture is correct.
Next time you are out with a group of riders, watch them from behind at walk and trot. Some tip left, some tip right, some lean forwards. Some hold their heads at an angle. Watch them at trot but also at the same time watch the feet in the stirrups. Eventually you can recognise a friend by the way he/she sits in the saddle.
Again back in the old days , it was typical for young ladies to learn to walk properly, that is upright with the spine aligned in the correct position. Some were told to practice by walking around the room balancing a book on the top of their heads.
If a rider gets into the habit of dropping a hip or slouching on the saddle then the muscles, tendons and ligaments come out of the ideal alignment. Then the back aches start.
We have a physio or two on the Forum - this subject is one on which they need to comment. All I know is that my back aches stem from sitting incorrectly in the saddle.
Before ever getting on a horse, the novice should be shown, with the help of diagrams and perhaps a mechanical horse how to sit properly. Most folks new to riding will then have to build up the muscles to support the correct posture. The older you are to start horse riding, the more difficult it becomes to learn to sit correctly because your muscles have already formed a pattern of use which maybe incorrect.
After every fall, the rider's back, hips and pelvis should be checked for misalignment. In any case the rider's back should be checked from time to time as matter of routine.
But I am no expert, I know only about the principles from bitter and painful experience - you need advice from an expert. Find a sports physio - they are the experts
PS I watched the manageress of an Indian restaurant walking across the floor the other week. As she walked she maintained an absolutely superb, straight, upright, posture. I asked if she rode horses - she didn't but as a young woman she had been taught to walk correctly. My guess is that she would have made a superb dressage rider. (It is a good job I am an old man)