I had to read this several times before posting. I see no benefit letting your horse stretch his head forward during any turns. All of the power for jumping comes from behind. Dressage exercises are the etudes that enable your horse to do everything better. Your horse should be able to jump low jumps (up to 3'0" verticals) from a walk. Stretching the neck forward isn't going to help that. Also, even if it looks
better, it won't help you with any speed.
A good hunter or jumper course is constructed so that the rider makes planned and fluid turns, and plans to make the turns efficiently and straighten in at least one stride before the next jump. Isn't it the challenge to stay round and rythmic so you don't jump flat and pull poles?
You know that engaging the hindquarters takes energy and effort, and horses are like us--all that exercise is HARD!!...so they will avoid it.
I have heard of letting your dressage horse stretch down after collection to avoid too much lactic acid buildup, and too give a release from tension.
Have you ever set up a course with just poles on the ground, walked it to plan, then ridden that? Try it and see if you agree with your trainer's advice, or not. Let us know.