I agree with the comments on high level dressage unfortunately. Dressage is basically training, and competing should be showing the level of training and communication between horse and rider, sadly I think the art of this, of truly training a horse well, communication with the horse and riding sympathetically has been overtaken by wanting results and quick 'fixes' have become more common.
I used to compete at lower levels and it was more about doing your best bit of schooling in front of someone you've paid to criticise you so you can work on that criticism and create a better partnership.
Racing and anything to do with jumping are more immediately dangerous, but I'm not sure these horses at top levels in dressage are always trained using the best methods unfortunately.
Although pp do have a point when they say that forcing a horse to do anything it doesn't want to is nigh on impossible, they weigh half a ton or more, have metal feet and can move sideways at speeds in excess of 40mph if they desire. It's not an argument you're going to win. It's done through cooperation.
Also that horses do perform these movements naturally, they are enhanced and taught on cue, but I've seen all types and breeds pull these moves in the field, under saddle when excited or spooking, playing etc, or just bimbling about.
Being honest though, when my horse started to get older and he retired, I did start to question a lot of things and felt conflicted. On the one hand is it fair to use animals in this way, but the other that we as humans, have developed and cultivated the relationship with these animals and I think as we have encouraged their dependency on us with breeding etc, we owe it to them to meet the different needs we've created in them that can't be met in the wild.