Gadgets gadgets gadgets. They're so popular now. I find it a bit sad that alot of coaches are now running off to find the nearest gadget to strap a head down and make a kid think their horse is 'on the bit'.
Sorry but I am 100% with kevinshorses on this. No you did not specifically ask for a 'critique' but this is in now way shape or form a critique as we have not seen you on the horse.
Generally, if I horse is throwing it's head up and down and hauling on the reins, unless he is sore, there's going to be an issue with the rider. Horses continuously throw new obstacles at their riders to try and escape pressure here there and everywhere, but you need to be a skilled enough rider to work a horse through these issues, without relying permanently on a 'gadget' to get you there. Sure everyone has to learn, but I don't think using gadgets for every issue is going to teach anyone much at all, what happened to putting in hard work and learning how to drive the hind end up to meet the hands to create a beautifull, soft relaxed horse? If your trainer is so set on you getting a martingale to fix the problem, I'd quit turning a blind eye and find someone who actually knows how to correct an issue such as this one, without having to use straps to hold the horses head down.
You can pull a horses head down, and that issue may be resolved, but the next one will surface quickly. If you rely on gadgets for every problem that arises, you may as well put your horse in a straight jacket, strap wheels to his hooves and get someone to tow him along while you enjoy riding a horse with a 'nice headset'.
Your comment that you love how a martingale gives your friend's horse a nice headset just proves that you are not aware of the concept of riding 'back to front', you are under the impression that a pretty, tucked in head is 'on the bit' or 'collected'. So SO not the case.
You don't need to take anyone's advice here. And I know you're probably going to go off like you did with Kevinshorses, but i think what needs to happen is that people on this forum need to stop thinking that they know everything there is to know and that if their horse isn't working exactly how they want it to, then OBVIOUSLY the horse is just being naughty. Get someone more experienced to ride your horse for a while and see what happens? If they know how to ride to a reasonable level of dressage/flatwork, they'll be able to combat the head movement by driving the hind legs under the horse, and you won't see the reefing occur again with that rider. We have to stop blaming the horse, take a step back and see that maybe our own riding is where the problem lies? So before you come back at me with teeth and claws out, just have a think... is your riding so perfect in every way that you couldn't possibly be inducing this behaviour?