What is wrong with riding a boarders horse in a lesson?
I often ride my instructors sale horses, and many of those horses ended up being sold within the barn, and are left available for students to ride in lessons. Some of the boarders bought upper level H/J prospects and left those horses available for more advanced students to ride in the owners place. Heck, even one of the barn owners horses is a beginners riding horse used often in lessons. It gives the horses exercise when the owners can't always get out to do so themselves.
A horse is a horse, and if you have the skill to ride them and learn to advance on them, then take it as it is. A horse to learn from, no matter if the barn owns it, the barn owner owns it, the trainer owns it, or if a boarder owns it.
It would have never been suggested if the owner was not okay with students riding their horses.
Good luck with your continued search. But do understand, if you want to ride, ride what ever you can get your hands on.