We didn't ban horses, we banned touristic carriage rides specifically because the few companies that operated treated their animals questionably. You can still see horses, just not ones that are getting abused.

And for what it's worth, even if you provided your animals with the greatest care (which they didn't), just putting them to work for 12 hours a day on a hot paved street was largely seen as abusive.

It might not have happened often that a horse fainted or died on the job, but when it did it caused enough of an uproar to pressure the people in charge to do something. As someone who has witnessed it, you really only want to see that once.