^^^ They don't do well in barrel or reining because the longer legs make it harder to do those sharp turns. Horses lower to the ground will naturally be able to make tighter turns and spins quicker because of their lower center of gravity. It's like asking why you don't drive a lifted jeep in NASCAR. Apples and Oranges.
That being said... To OP, I'm a tall horse girl myself. I like them 16hh or bigger. My gelding is 16.2 and I love him. He is very easy for me to ride. A little hard to get up on but I just have to keep in shape so I am able to mount up as easily as possible. Good motivation to stay fit.
I didn't have a single lesson before I got my first horse. Mine was an OTTB that was given to me by my grandfather. However, my father-in-law, has ridden his whole life and was there to give him a test ride in the round pen before I took off on him. I've learned as I've gone along. Comic has learned right along with me. "A lot of wet saddle blankets" is what my FIL told me. We took a few lessons about a year after I got him but other than that it's just been trial and error and listening to advice from others.
Be patient as you try to find the right horse. I think you could start looking now if you wanted. Ask your trainer to help you look at horses. I think it's good for a person to learn with the horse they own. Nothing wrong with lesson horses it's just a different kind of bond with your own. But take your time and look at lots of options. When you find the "right" horse, that you and your trainer agree is the one, buy it. Doesn't matter if you are 2 weeks or 2 years into lessons IMO.
Riding: The art of keeping a horse between you and the ground. ~Author Unknown