Personally I think it's a case by case thing, not breed by breed.
I have two thoroughbreds currently, but in the past I've had others.
My first thoroughbred HATED a normal snaffle, but went fine in either a mullenmouth or a hackamore.
Mitch, well he head tosses enough to knock somebody out in a normal snaffle, so he is in a fixed ring french link.
DJ, she does just fine in a plain old loose ring snaffle. No problems at all.
Conversely, a friend of mine with a HUGE dutch warmblood, has to use at least a double jointed bit, if she uses a single jointed bit, he rears.. And at 17.2hh with a head the size of my torso, rearing is not fun. But put him in a double jointed bit, or a waterford etc, and he goes just fine.
Also, a lot of show horses in NZ are thoroughbreds, it's the easiest breed to come by, most of them show in double bridles with no issue at all, even the teeny little petite ones. Also, a lot of the welsh ponies and arabs show in double bridles with no problems.
So personally I think it depends on the horse, not the breed