I would venture to suggest it is impossible to train conventionally without negative reinforcement. The only people who would argue they do avoid use of negative reinforcement altogether would be some of the pure positive reinforcement / clicker trainers, and even then I imagine it would take a high degree of skill and dedication.
I think you may be using the term 'negative reinforcement' to mean the same as 'punishment' rather than 'pressure and release'. Apologies if this is not the case and I have misrepresented you. Clarification may be helpful for those who hear the term 'negative reinforcement' and think it refers to reprimand or correction.
I can't speak for ycbm, but your statement pushed a button. Punishment doesn't necessarily mean a "damn good thrashing" or indeed anything violent or even painful or scary. A horse that learns to avoid bitter tasting plants because the bitterness is unpleasant is punished for its poor choice of nibbles. Punishment spans a whole spectrum of severity. Good horsepeople know how far they need to go and will naturally use the mildest form of punishment that is effective. To do otherwise consistently shows a lack of skill and awareness. Seeing someone administer a "damn good thrashing" would almost certain indicate they weren't a good horseman or woman.
No, but there are lots of unacceptable behaviours that can be stopped for the good of both the handler and the horse. Of course, what is "unacceptable" is to some extent in the eye of the human beholder. Some people find a horse rubbing an itchy face against them completely unacceptable; others don't. There isn't a hard cut definition of rudeness or "bad behaviour", just conventions.
Behaviour can be modified for everyone's benefit. Limits should be taught when it comes to natural behaviours that are potentially dangerous. Of course, the consequences for behaviour - good or bad - should come immediately otherwise the horse is unlikely to know what it is being punished or rewarded for. Thirty seconds is effectively an eternity for a horse. Even three seconds is liable to be too slow, given the horse has probably moved on to its next thought by then.
Great if you see the kick coming in good time. Unpleasantness can be avoided altogether with enough awareness. The problem with kicking is that if you are unlucky to actually receive one, or a near miss that shocks you enough, it may be too late to punish the action effectively by the time you have gathered your thoughts. Doing so many seconds after the event is almost certainly not going to help and may make matters a worse. If you get kicked and don't respond instantly (best) or within a second or two, you should keep your counsel and reflect on what you could do better next time.
I routinely punish coltish nibbling and mouthiness by sticking a finger or knuckle into the palate of the mouth for a few seconds, something the horse finds unpleasant enough for it to be an effective punishment. It's devoid of emotional content, the horse thinks he has inflicted it on himself rather than perceiving it as something I have done to him, and I don't need to do it more than a handful(!) of times to completely stop the behaviour for good. Much better than an angry reprimand or bopping it on the nose, I think.
Almost all the horses I encountered in Malaysia when I accompanied a vet on his rounds showed some degree of headshyness. I assume it was related to standard handling practices. Wish I'd asked about this.
It's easy to say (or imagine) one would have reacted differently in such a situation. Of course one hopes one would, but if it came down to it I doubt I could put hand on heart and swear I would. Actually, I can think of situations in my own life where to make a fuss would have been counter-productive because it would have led to me losing all opportunity to improve the lot of the horse or horses involved.
Effectively, that is punishment, even if you would rather not call it that.
With respect, Seville, it's not always that simple.
