If I can walk in NY, a horse can walk in NY. If these animals are not being properly cared for by their caretakers, that is one thing, but the idea that a horse drawn carriage or being around people and cars is somehow animal cruelty I personally find quite ridiculous. And the reality wake up call should be pretty apparent on what the fate is of animals that have no function or utility in society... many existing ones will be put down and eventually simply less will exist.
And I'm quite sure the entire point is that it is antiquated, as a way to remember how things were and share that bit of history, culture, and tradition of our ancestors before us and give a sense of continuity that all that is old is not forgotten and gone forever. Same thing with Buckingham Palace and the silly poofy hats, it is antiquated by design.
Lastly, in the current climate of "going green", what can be more environmentally friendly and provide a touch of nature than a horse drawn carriage in an otherwise sterile concrete environment with polluting steel carriages burning fossil fuels rather than grass as the alternative?
For those that support banning keeping such animals for our amusement and pleasure, don't be too surprised when PETA then uses the same logic to ban companion animals from captivity and human slavery.