Maintaining a lesson string is an expensive task and time-consuming task. It’s a much more feasible business model to cater to full and part training clients. It doesn’t take much to keep a hunter tuned up as a lesson horse, and leases are the name of the game in that world.
Most average dressage horses need constant correct training to keep them up to par. Students slowly ruin good horses. For every lesson they are ridden in, they will need a ride by a trainer. That’s time that the trainer could be using to ride a client horse and get PAID. If I’m a trainer and I buy a horse that has a good attitude and is fancy/athletic enough to do 3rd level, I am flipping it for $$$$ to the first amateur with an open checkbook that walks in my barn door. Maybe teach a few lessons on it, or let a working student ride it/show it, but the goal is to get it sold and keep it advancing and competitive while it is for sale.
A lesson horse will be a horse that is not really competitive or sellable for some reason, usually not-so-nice gaits or aged out of competitiveness. Not-so-nice gaits mean extra training to get them rhythmic and cadenced so the students can sit on the damn thing. An aged horse will need maintenance: shoes, injections, special feed, supplements, etc. It takes a lot to keep these older horses going.
I used to be frustrated by this as well. But after working for more than a few trainers and seeing how tight the margins are for a professional in this sport, I totally get it. I know of a few trainers who maintain 2-3 lesson horses, but they live in the boonies, and those horses all fit into one of the above categories I mentioned.
I’ve taken group lessons in Europe. The trainer insisted that we maintain one horse length between us at all time, so she didn’t ever have to turn her back on a single student. You really learn how to dictate rhythm and how to truly half-halt and balance your horse every stride in school riding. We’d split up for flying changes and pi/pa work, but other than that - it was single file around the school.