Predators that would go after a deer could also be harmful to a human. Even if it was native.
But on the other hand, a few lions here and there would certainly make NY an even more interesting place.
More seriously now:
Putting deer (and other critters as well) on birth control is nothing new, and is actually a rather tried-and-true population control method. But, why not do something that would seem to be effective? Surely, if we can administer birth control, we can administer poison? Or why not just hunt them (you may ask)?
Well, poison is just bad PR. Also, you run into the problem where the poison might stay in the environment, or might be ingested by something other than deer (like you your pet dog). Birth control (which in the case on deer is typically an immuno-contraceptive) is a much more host-specific method.
Onto hunting. The simple answer is that in most situations where contraceptives are used, the deer population is in/near an urban location where it wouldn't be a great idea to start shooting off guns. As anyone who has ever been outside on the east coast knows, deer really really like the suburban environment (garden fresh food + no predators).
The longer answer is that, as non-intuitive as it might seem, hunting doesn't really reduce deer populations*. This is because deer populations tend to be resource limited, and because most hunters tend to go after male deer (bucks). Removing a buck from the population doesn't reduce the size of the next generation, because there are still plenty of bucks to impregnate the does. Indeed, hunting does doesn't even have as large of an effect as you think it might, because you are also increasing the amount of resources available to the remaining does. This increases the survivorship of the offspring of the does that were not hunted.
Sterilization, however, directly reduces the population size of the next generation. Furthermore, because the sterile does are still consuming food, they are taking resources away from the fertile does, thus making their offspring potentially less nourished. Its rather sinister actually.
*I am talking about recreational hunting here. When deer populations are actually managed using hunting, professional sharpshooters are employed. They have a much higher success rate, and can be directed to target does. This is actually the most cost effective method of controlling deer in large areas where sterilization would probably fail.
