I do kind of see where you are coming from but, knowing horses, I do know that it takes a horse very little time to settle into a new home with new people. Yes it takes time to build up a trusting relationship between horse and rider but a horse will generally be happy enough by the end of a week in a new place. Some take a bit longer, some a bit less, some react more, some react less. Horses do work up bonds of attachment but not in anything like the same way as people do.
I have a problem with people who get rid of an old horse simply because it is old but selling a horse to someone who can use it properly when you no longer can seems to make sense to me, most horses love thir work and pine when left in a feild doing nothing, it would be a bit like telling someone whos job has been changed that, as they can no longer do that job they cannot do anything at all and so have to go and stand in a corner for the rest of their working life!
Many dogs are passed from home to home for various reasons, working, showing, breeding etc all lead to regular home changes, generally these dogs seem to be happy individuals who are perfectly capable of forming the usual doggy/person attachments.
TBH I think this is a bit like the "sheep for meat" argument, if you are not in the middle of the situation and experienced it is difficult to have an objective viewpoint rather than a more emotional one. Rightly or wrongly I do see horses as animals not people, I have a working relationship with, I form attachments to them but I do not lose sight of the reality of the relationship - the same as I was perfectly straight about the relationship I had with the beef cows I helped raise and the dairy cattle I showed when I was a child, for example.