Some of the answers here are advocating keeping the horses in the park and on Central Park South. It's important to note that many of the accidents we know about (there probably are numerous others that aren't reported) have occurred in the park and on CPS--57% in 2006-07 and 31% between 1994-2007.
If you talk with the artists who draws portraits of tourists near the hack line, they will tell you about how they have seen many incidents that are not reported whether traffic accidents or horses collapsing in the summer from heat exhaustion. The accidents we know about are only the tip of the iceberg.
Restricting the horses to the Park and CPS is still dangerous, and still subjects the horses to extreme conditions that can't be changed through regulations--extreme weather, hard pavement, loud noises and other phenomena that make them spook in park, traveling back and forth to stables through heavy traffic, confined between shafts of carriage for long hours and in small stalls in warehouses. The very high turnover of horses in the industry speaks to this hard life they are subjected to. After their four years, they are sent to New Holland auctions, where they are most often purchased by killer buyers, who ship them to slaughter houses (for European consumers, who like to eat horse meat), these days in Mexico as they have been shut down in the U.S. It has been documented in industry publications that NYC carriage industry is the main supplier of carriage horses to the slaughter industry.
These facts seem incredible, because they are so awful. But they are true. If the carriage horse industry retired its horses to farms outside the city, there should be hundreds of retired horses we should be able to go see.
I grew up with horses -- my grandmother had a farm. The horses I see on the hack lines look tired, often frightened, and generally maladjusted. It is no wonder that they routinely bolt and cause traffic accidents. We've seen 4 this year, I believe? And I've personally seen the horses treated quite poorly -- out on dangerous, slushy streets, discplined too harshly by their drivers, and so on.
Compromise just won't work in this industry -- The horses don't belong in the city in 2007. I honestly don't understand why tourists think of these rides as romantic, or iconic, but perhaps they could walk. Young boys used as chimney sweepers were iconic in NYC too in the 19th century (and no doubt cute in their day). Should we bring them back too, to fulfil our sentimental fantasies?