I would speak up and calmly but firmly state how the horse would prefer to be handled, and if that doesn’t work, ask the veterinarian directly either to have a different tech hold the horse, or if they’re comfortable with you doing it.
I’m a vet tech, in small animal not large, but grew up riding horses. Unfortunately, most technician schools barely cover horses or any other large animal for that matter. My school had one class on all large animals, of which horses took up about 3 weeks. We physically got a chance to handle horses three times and that was it. People who don’t know horses do not learn nearly enough with that- and of course they only handled elderly, dead-broke, everyone-safe horses, and don’t get experience with horses who vary from that at all.
So you’ve got techs coming right out of school but with almost no horse experience. If your practice has assistants, they probably have no training/education whatsoever, as they don’t need a license or any schooling. The field is understaffed literally everywhere and it is near impossible to hire anyone. Most practices just have to take what they can get in terms of techs and assistants, and a lot of them are bad.
Veterinary practice insurance does usually require that only paid staff are allowed to handle animals, not the owners, just for liability reasons. However, it varies practice to practice on how strict that is. Many vets are willing to grab another tech who may be able to handle better, or to let the owner hold, especially if it’s a pretty easy going animal with a low risk of hurting anyone, or a ridiculously aggressive or stressed out animal that unless the owner restrains, nothing is getting done.
You can also talk to your vet and ask them to put a note in his file that he likes being held on a loose lead, or that you’d rather he wasn’t held by that tech. My coworkers all listen to notes like that and no one is offended that certain patients do better with certain people/handling styles.