I have taught several guys to fly. One had a PT-40 and another had a Goldberg Eagle II. Both planes made pilots out of the owners pretty quick. We didn't have any buddy boxes. We just passed the transmitter back and forth. That worked fine. I have also taught on sport models like a Scooter racer. And A Smith Mini Plane. Pretty much whatever plane someone already owned was used for a trainer. Turn the controls down and about any plane can be a trainer.
When I started flying back in 1978 the guys at the field recommended a Das Little Stik for my trainer. So I bought a kit (no Arfs then) and in a week or so had a plane. When I went to the field my "instructor" mostly flew my plane for me. As soon as I had the basic concept down I would go during the week to a nearby park and fly by myself. The plane got a little beat up but was always repairable. I learned to fly with a plane that today would only be suggested for a second plane.
So a trainer can be something other than a flat bottomed airfoil model. It really comes down to the instructor and his ability to teach. But there is absolutely nothing wrong with a dedicated trainer type model. I like flying trainers and have a Tower Trainer I fly all the time. It is the plane I take to the field most often.