The vast majority of kids in the early to mid teen years are simply at a stage in their development where they are more open to learning from a respected adult OTHER THAN a parent. Just the natural course of things. Perhaps if your child is, as mentioned already, more compliant than most they might pay a bit more attention, but a trained professional is worth it.
My kids could run a tractor (go cart, golf car, boat) when they were 8 or 9. That sort of skill is not really what one is concerned about when teaching your child to drive on the roads today. Half the battle is defensive driving, and certainly not going backward and forward!
Paying the extra money to better prepare your child, when honestly their life is on the line, is the responsible thing to do, IMHO. If you try to justify it with "oh, some paid classes aren't good" -- come on, that seems to be a cop out. Find a good driver education program!