I was going to say something similar. I can see the appeal in buying a made horse and being a passenger. Indeed if you are primarily in lessons or even a lease, the situation creates that orientation to a horse. No one wants you to be training new things into a lesson horse!
But if you want to ride effectively at higher levels you need to make that conceptual jump from being a student/passenger to being part of the training team. Every time you get on a horse you are teaching him for better or worse. With your own horse that you are the primary rider you will obviously still need some very directed instruction in how to train and advance your horse. But it’s you who will have the biggest impact on his way of going.
The dividing line between student and owner, or beginner and intermediate/advanced is the ability to be a major part of your horses daily training. Rather than think training is something you leave to the professionals and you only need to work on yourself.