According to "Notice of Adoption of an Amendment to Article 161 of the New York City Health Code" the argument for off-leash areas stems from this:
"[Department of Parks and Recreation] has informed the [Department of Health] that this policy was implemented at the request of dog owners who had no alternative place to exercise their dogs."
Why the demand for off-leash in Central Park? I guess when you bring your dog to an enclosed dog run you actually have to watch it AND clean up after it.
(Click to Enlarge Map)