If you are looking for a scientific study showing a horse responds to to XYZ 23% better in a bit that bitless, no. Nor is there any on the other side, altho some manufacturers try to claim there is.
The problem is too many variables. There are a wide variety of bits and bitless. One horse will do well in a french link snaffle and hate a standard link. My mare preferred the standard link to the french link. Some like loose rings, others hate them. My mare seems to work well in a western curb, while others do not.
Then you have the REAL important stuff - how they are used. I could make a horse hate snaffles by how I rode them. Same with curb bits. Same with bitless. You cannot isolate the effect of the rider, so you would need a huge sample size to even out rider use. And what standard do you use? A horse may act relaxed because it is confident it knows what its rider wants, or because it knows it can ignore the rider.
I think what you WILL find is that almost no horses anywhere are trained to a high level without using a bit for training.
The flip side is also important - bits do NOT work on pain. There are far too many horses who act happy with a bit. Bits do use pain as a backup for control. If the horse is refusing to listen, then most bits allow the rider to cause more pain than most bitless bridles. However, that is a GOOD thing. Anyone who has ever been on a bolting horse heading for a highway or steep ravine understands that it is better to darn near rip your horse's head off than to have one run into a car doing 60 mph, or off a 10' drop concealed in bushes. The truth is that horses NEED the rider to take control.
Horses are emotional creatures. When their emotions are running strong - fear, competitiveness, etc - then you need to get past the emotion controlling them. Think of it as cutting thru background noise. In a mountain meadow, you can whisper to your companion. On a flight line of jet engines, shouting may not be enough. A 'harsh' bit may make it possible to 'shout' loud enough to save the horse's life, and yours.
All of these factor in to the decision to use a bit or not. The biggest fallacy pushed by most bitless fans is that bits cause pain. On rare occasions - racing to death, for example - then yes, they cause pain and save lives. But lots of us have horses who strongly resent pain, yet who will take a bit without any problem.
Does this look like a horse in pain? She has chosen to pull the rein tight while looking, but do you see pain? I don't...