I don't agree at all with using animals for sport, but it does make me wonder why this video surfaced now, and who the source was. Bearing in mind it's caused the withdrawal of someone who was poised to be the most accomplished woman Olympian ever, there are vested interests outside of animal welfare that make me suspect motivation.
Also, hundreds of horses get shot dead on racetracks, all the time, when they break their legs, because they're so hyper-specialised, they have the ability to run fast, but fixing them is too costly in time and effort, so they're "put out of their misery"... Never mind the gambling-centred racing industry that hyper-bred these animals expressly for entertaining us. And then worked them literally to an early, if relatively sudden death.
Just, don't use animals. It is so very, very straightforward when you set yourself that basic goal, and no amount of "oh it's an industry! Oh it keeps farriers in work! Oh it brings us medals to boast about, oh it keeps fox hounds happy, oh it clears the fields of pests, oh it's what the traditional King's Guard sits on.." It's all the same continuum of species-specific assumed superiority, and greed.
Horses in the wild don't dance like hot pricks. We have made them dance like pricks, because it pleases us. Distasteful, archaic, and exclusive, it can all get to fuck, whipping or no whipping.