dz,
We had/have a similar debate here in Michigan. There are "Metro Parks" in the southeastern area of the state similar to state parks. Most, if not all, have deer overpopulation problems due to public feeding (not allowed but people do it anyway) and no hunting policies. I attended a public meeting about 2 years ago on the issue concerning a nearby park. The anti's espoused the typical mantra that you have stated. They even went so far as to suggest tranquilizing deer and surgically sterilizing them. One woman said she did not see a problem even after pictures showing an obvious browse line and loss of plant species in the park.
After much debate, although I believe officials had already made up their minds, the herd was to be "professionally culled". At first this meant sharpshooters but then they decided to train and equip park personnel to do it. To this day they are still doing it but in some of the other parks they have controlled hunts. This still baffles me as most of us were willing to pay the park to hunt and would donate the meat to local charities. But then that would mean dead deer and feeding the homeless. What were we thinking!?