The ONLY time I have seen a running braid was on a fat little short stirrup pony at a low rated show. Maybe other areas are different, or schooling shows, but at rated shows, it is expected that the mane is pulled, and usually braided. It shows respect to the judge to groom and trim the horse to make it look neat and clean. A braided mane (proper hunter braids) must have a pulled mane. It shows of the horses neck as well as showing respect.
As for why do it not for shows? My old horse had the *thinnest* mane I have ever seen, it was 2 hairs thick! I cheater combed his hair to make it look slightly thicker, it was ugly long. It is a requirement to pull manes at our barn (with a few exceptions). IMHO it looks neater, is less work, and a long mane gets in the way riding. But if you like a long mane and don't show hunters, there is nothing wrong with a long thick mne :-D TO each their own