Yes it can hurt the horse. That's the whole point. And of using a bit, whip, spurs etc... prong or shock collar on a dog...
BUT when a horse(or dog) is taught well *first*, without punishment, and the handler is knowledgeable & skilled with the use of tools like that, then it doesn't NEED to be painful generally. The animal can then learn how to *avoid* these things becoming painful, as
explained. It can then be reserved for 'emergency' use, when it may be necessary to use force for control.
Eg. I use predominantly positive reinforcement/reward based training with my animals & seldom find punishment necessary(think it's generally best avoided where possible). But when the 'chips are down' & my normally obedient dogs are loose in the bush when a kangaroo goes past & their little hunting brains no longer register anything but instinctive chase, they are wearing e-collars(prior trained to understand them) and the pain of a light static shock is needed to stop them in their tracks & get them coming back to me - and rewarded for doing so.
*I do think a chain over a horse's sensitive nose or lip is a bit of an exception to the above tho - think that's generally painful, or at least quite uncomfortable - & so would not use that.
**Note again the; when a horse(or dog) is taught well *first*, without punishment, and the handler is knowledgeable & skilled... I believe it's a 'prerequisite' that the animal AND handler be well enough educated & skilled BEFORE using tools such as these, to be 'humane'.
I dont agree with just using stud chains on young studs for eg for control without first teaching them how TO behave. I also don't believe it's humane for an unknowledgeable or unskilled person to be given those tools to use on an animal.