Historically, “leverage” bits (in the military or hunting field, usually the double bridle) were considered the complete and appropriate equipment for a horse with an educated or “made” mouth. One who kindly accepts flexion of jaw and poll, yields to pressure, and is schooled to the indirect rein (neck reining). Such a horse would not “pull,” and a light hand on the double bridle easily kept him balanced, on the flat or over fences. Leverage bits most certainly ARE good for turning, you can ride the horse with one hand! (Watch a reiner sometime!) 
This was called a “finished” horse, and to ride him a rider was expected to have “good” hands. Meaning, you weren’t using them to balance, water-ski or otherwise keep yourself in the saddle. On the flat or even in saddle-seat, the snaffle was used to raise the neck and withers and the curb for relaxed flexion. Your trainer allowing you to use a double bridle was up there with getting your toe shoes in ballet–it meant you had reached a certain milestone in skill and preparation.
Such was the state of the art through the end of the 1960’s. You will see MANY pictures of hunters, jumpers, equitation horses, foxhunters, and eventers using this as the “default” combination of bits under the “finished” or expert rider.
In the 70’s the Pelham, particularly rubber, became popular because it was a little more “forgiving”–one didn’t need to have “as good” hands as for a double, but still had more “brakes” and control than in a snaffle. Foxhunters found it a good compromise, and pony and children’s hunters found them very efficacious.
The MOST popular bit used in riding schools for beginners was the Kimberwicke, because it allowed beginners to have decent control before they had a truly independent seat; having no shank, it is not truly a “leverage” bit.
The “broken curb,” "broken “Pelham,” “Tom Thumb,” etc. are not “leverage” bits at all, because when the “leverage” would be applied it breaks in the middle. If anything, all these bits accomplish is to provide turning stability due to the full cheek coverage afforded; functionally, they are snaffles. They can also have an enhanced “nutcracker” effect which can be problematical.
Traditionally, it was considered an exercise for a very, very advanced horseman to ride a horse through all his paces, in the ring and across country, in a plain snaffle. Such a horse was exceptionally educated for sensitivity, and the rider also had a very sophisticated knowledge of obtaining balance and flexion. This did not apply to the racecourse, where the horse leaned on the snaffle for balance and was “rated” by main force.
With the importation of Warmbloods as the sporthorse of choice in the 80’s and the “Eurocentrism” that followed, riding styles changed to more “behind the motion” and “driving seat” and somehow or other the notion that a snaffle alone was a “kinder” bit for beginenrs mutated from the dressage show arena (a very controlled environment) to being elevated to general use. All of a sudden you “weren’t quite nice” if you didn’t ride your horse in a snaffle–never mind that he frequently wasn’t educated to flex and yield and so you were getting run away with in the the hunting field, jumping chancy and flat on XC, and people with all good intentions were teaching their horses to pull by hanging onto the reins from an incomplete teaching about “contact.”
Which brings us to where we are today.
Personally, I’d rather ride ANY horse with my fingers and weight than with my arms and muscles. If you had to ride him all day it’s a lot less fatiguing. A horse who comes easily back to your hand upon the nearly invisible aid of sitting up a little straighter and closing your leg, who can invisibly add a stride before a fence, who is listening and not resisting, is an awful lot safer than one who is engaged in a wrestling match for balance, at speed over terrain. And riding him is a PLEASURE, which combined with safety is a lot more important to ME than being “politically correct.” My best advice is to ride in what works best for YOUR horse, given your skills, if it’s allowed by the rules in competition. Nothing says you must use nothing but “allowed” bits at home, either. There are many horses in the world who just “don’t get” flexion in a plain, broken snaffle. Years ago these things were understood very well.
