I have successfully retrained contact resistant lesson horses to be light and responsive to just the snaffle (Wellep single & double-jointed snaffle, JP Korsteel Dr. Bristol D-ring snaffle with sewn on bit loops and the Fager titanium Bianca, double-jointed eggbutt snaffle with a titanium roller on the center dog-bone). After succeeding in getting mostly light and responsive contact and obedience with these snaffles I then changed to using the double bridle (Fager Victoria titanium Mullen mouth Weymouth curb and the Fager Alicia titanium eggbutt 3-piece bridoon whose mouthpiece is similar the the Fager Bianca mentioned above).
I got further refinement from these lesson horses. They further relaxed their poll and jaw muscles and their tongues became more responsive to my lighter hand aids. Two of them ended up giving me clearer and steadier contact when I kept light contact with just the curb rein and their poll and lower jaw muscles relaxed even more while they voluntarily reached out for contact with the bit.
Using the double bridle for me resulted in previously resistant horses learning to use the double bridle bits to communicate with me, which also contributed to the increased relaxation and responsiveness of their poll, lower jaw and tongue muscles.
With just the snaffle bit I can get light control and the horse can tell me somethings at a really basic level. With the double bridle I get light control with which I can give rein aids with a small properly timed twitch of the appropriate finger while on contact, and the horse can give me ever increasing subtle poll, tongue and lower jaw responses and both the horse and I get into an interesting discussion of how to do whatever I ask for RIGHT, and believe me the horses can tell if it is RIGHT (their soft snorts of contentment and accomplishment are music to my ears).
Any rider can use any piece of tack with brutality resulting in very unhappy, totally miserable and resentful horses.
Good riders learn to use tack with finesse and delicacy, limiting harsh aids for total EMERGENCIES where the horse has to respond immediately for reasons of safety. The rest of the time we aim for engaging the horse while the horse stays mentally relaxed and physically responsive to our aids.
Believe me or not, the added refinement from using a double bridle with light, responsive and educated hands blows just using the snaffle alone out of the water. Sort of like the difference between discussing a scientific subject with a very intelligent child in the 5th grade and talking to a PhD in that subject.