We'll have to agree to disagree. I don't agree with any gadget that works on the front end of a horse and I'm not a big fan of the Pessoa lunging system either [which works on the whole horse]. I use side reins while lunging because it gives the horse a consistent contact to work into, much the same as having a rider on their back. The horse that is correctly mouthed and correctly trained naturally comes into a headset without the need for gadgets.
My learning about bits and spurs is from a dressage standpoint where they ARE used to refine. Used correctly, the bit and the spur are actually completely unnecessary. Take any [CORRECTLY trained] Grand Prix horse, and its rider, and remove saddle, bridle and spurs, and ask them to ride a dressage test. The horse SHOULD be ridden so much off seat and body position that they can ride any movement at any level, through any sequence, with no issues. They compete with double bridle and spurs for REFINEMENT. Of course there's bad dressage at every level but the spur is so that you can refine your leg aids. With a bare heel, you have to move your leg more [therefore giving more visible aids] to be clear to your horse what it is exactly that you want. The point of dressage is to appear as if there is a telepathic connection between horse and rider.
Hence, we refine. And we refine. And we refine. The same with the double bridle - refinement. You can give two aids at once with a double bridle on, with the two different mouthpieces. The bit and bradoon is potentially an incredibly harsh combination in the wrong hands, but used correctly, on a horse that is properly trained, it is a refinement tool. The horse that goes in a double bridle should go equally softly in a snaffle.
The spur is not for digging into the horse's side. It's for TOUCHING the horse's side with an accurate, refined aid. The horse ridden with a spur SHOULD be equally soft and responsive with a bare heel.
I have to wonder how much your trainer knows if they are encouraging the use of gadgets to be honest... I know MINE actively DIScourages martingales of any kind, harsh bits [she only puts up with the fact that I ride in a pelham for jumping because she knows I -can- ride my horse jumping and XC in a snaffle, I just feel safer with 'emergency brakes'], any sort of gadget. The most she will allow in noseband-land is a flash. Cranks are unacceptable, grackles only ok in jumping and even then ONLY because they restrict the nostrils less than a cavesson or a flash. She is a professional competitor and trainer as well as a riding coach. She teaches English, the RIGHT way.
A CLEAN SLATE FOR THE FUTURE