Well, I'm 19, and I consider myself a capable enough rider to teach beginners. But I would never dream of teaching advanced horsemanship techniques - even even if I can do them myself.
I personally found easier to learn on one mount, instead of changing all the time. Once everything has been partially mastered, or at least understood, then it is good to change horses. A confident and skilled rider will be able to handle all types of horses - but it helped me so much to learn on one solid horse. I think beginners need a been there - done that mount, that are not soured by lessons or lazy. I know that's hard to find in schooling ponies these days.
When I was young, I took lessons on schooling ponies, a different one each time. I learned, but it was difficult to adjust myself to each horse's gait and movements. It was slow-going. Then I started riding Shorty, and teaching myself to ride bareback, and it was so comforting to have him on my side. If I fell, he stopped and waited for me to jump back on. If I asked him wrong, he waited for me to ask him correctly. He moved out when I wanted him to, and slowed when I was ready to stop. He was a great horse. He made me feel like I could do anything. RIP, good horse.
Anyway, I'm going off on a tangent. To answer your question, I do think it's better to learn on one horse rather than switching all the time. Perhaps there is a horse you could consider leasing, so that he/she might always be available for your lessons? I think that the right horse builds a TON of confidence, and when you're confident in your abilities, you will grow so much more quickly.