Many people don't think that they would need rider strength and conditioning. I used to think that too. I used to think I was strong and fit - I rode 3 horses a day, mucked out 20 stables and did about 40 haynets, as well as cycling to the yard and playing sports. But once I stopped riding and went to the gym, my riding suddenly improved.
Riders and sports people tend to think about the skill required for their sport and not the capacity. They focus on the exact technique and position, rather than wondering whether our body (muscles, joints, lungs) can physically do it. Strength and conditioning is about training you to be a better athlete, so that you are more able to do your sporting skill.
For example, if you have good balance, leg and mid-back muscle strength, holding the correct position when riding will be much easier. This will then allow you to focus on your horse and help him improve, rather than being a limiting factor.
However, its the ability to do this properly, with all the subtleties of a professional rider which makes the biggest difference. This is the key difference between having a 1:1 session and just "going through the motions". Most people are a bit tired after their first Rider S&C session because they have had to think so hard at keeping their body straight and in balance, which they haven't done in a while. Even very good riders who have the skill struggle with the exercises because they don't have the capacity.
I hope you have found this distinction helpful, please email if you have any questions about this or if you would like to see if there are any ways you could be improving your capacity to improve your riding skill.