I have jumped a lot over the years, but have just started XC schooling with my horse. My horse hasn't had a professional trainer on his back, but I take and have taken a boat load of lessons. Basically my point is is that we are learning together at this point in time.
Anyways, wit that being said, my horse has saved my butt on numerous occasions. IMO, the hardest part about learning to jump is getting the correct distance to the fence. And by correct distance I mean what feels the best for horse and rider. When I ask my horse to extend or compress his stride in order to make a nice distance to a fence, he listens to me. Granted, I don't always do the right thing (sometimes I ask for an extra stride and I probably shouldn't have), but thats why we take lessons, right?! Anyways, there have been times, both xc schooling and jumping in the arena, where three strides out I'm not sure what to do and he figures it out. It doesn't always look or feel the best, but he does figure it out. And we are improving! So I guess I would have to disagree with that statement as well. It is definitely a team effort from both horse and rider.