Wagtail
Horse servant
You might. I certainly wouldn't. I really don't think it hurts. If it did the vast majority of horses wouldn't stand there and let you do it. I think some just find it an odd sensation and get a bee in their bonnet about it. Twitch is the perfect solution for these situations.
Have you ever been waxed? It's hardly agonising. I don't suppose there's much difference really.
Of course it hurts! Many horses are very stoical and just put up with it because they have been trained to. It is completely unnecessary and why anyone would want to twitch a horse in order to do it is beyond me. Twitching is for essential things. You CAN get just as good results with humane methods. It just takes a lot of practice.