Dr Cooks published papers aren't going to convince me my eyes are liars.
Yes, a bit can hurt a horse. And yes, if your horse is panicking and not responding to cues, a sidepull halter can take the hair off of your horse's face - and I suspect that involves some pain too. I know because I've done it. And if you pull hard enough on Dr Cook's bitless bridle, you will put a lot of pressure on the jaw. I'd bet that can hurt as well.
I would bet the bit I used today hurt Mia several times. She was having a "You're not the boss of me!" day. At one point, she nearly backed into a large cactus, and THAT would have hurt her too. Not to mention what would have happened if she ran out onto a road where the speed limit is 50 mph.
When she was about to ram the cactus, I swatted her rump HARD with a leather strap. She didn't like it, but she would have liked the cactus spines stuck in her butt a lot less! We did some tight 360s, too. When she wanted to canter (or faster), we stopped. Didn't need a pulley stop, but I wasn't being gentle on the reins.
I wish I could have been. I prefer riding with slack in the reins. Heck, I'd prefer riding with a sidepull halter - but a horse who refuses to listen to her rider can kill herself and her rider. And I don't want to die.
After about 10-15 minutes of near constant fussing, she decided to give in. For the rest of the ride, turns were based on leg cues. Back to the arena, we cantered a couple of laps and then stopped from the canter. A complete and quick stop. No sliding, of course, but she didn't waste time, stopped with her feet squared and didn't move until asked.
After dismounting, I removed the halter, held my hand next to her face, and she rubbed against my hand for several minutes. No, there was no blood on the bit and no sign of injury. I needed to wait for some others, and she stood relaxed next to me, nose at my hip.
Yes, bits can cause pain. So can bitless bridles. Neither is intended to cause pain. But horses get a vote in things, and a horse who votes to endanger her rider needs consequences that will change the vote. Nevzorov's solution is to stop riding. But my horses get bored with a 70' corral, and I don't want to pay $20/bale of hay to feed large pets. Heck, my wife didn't feel like riding today, so she walked our little mustang on a lead line with us (my daughter & I and our horses).
But while there was nothing mild or kind about my hands during part of the ride today, at the end my horse enjoyed her face rubs, enjoyed getting her mane brushed and seemed to enjoy hanging out with me as we waited for the others. I played with her bangs and scrapped some goop out of the corner of her eye with my fingernail. And unlike Nevzorov's horse, she didn't try to bite me...