I live on Long Island and I think that people will still go to the city as tourists without the carriage rides. There are so many different things to see and do in the Big Apple. I mean you could go see shows on Broadway, you could go to all kinds of famous restaurants, there are so many different attractions as well. People go to see the Empire State Building, go to Central Park, and even just to experience what it feels like to go there. Don't forget about Madison Square Garden either, the events that tale place there attract numerous people too. Furthermore, there are so many museums like the Museum of Natural History, the Bodies Exhibit, the Museum of Modern Art, and even the Metropolitan Museum of Art. There is the statue of liberty and Ellis Island. You also can't forget about the shopping!
Sure, some people enjoy riding on the horse and carriage, put all in all, I don't think that many people travel all the way to New York just to ride a horse carriage.