There's a difference between exterminating pests (rats, voles, raccoons, or any animal which is causing severe problems) and hunting an animal to near extinction (California Condor, Grey Wolf in Yellowstone). I agree that the animals are a problem, but the slaughter does need to be controlled.
apples, oranges
99.9999% of the argument the "animal lovers" have against deer hunting comes from a position of pure ignorance, and is based more on emotion than anything else.
Whitetail deer (at least in the SE US) have virtually no natural predator other than man. In our area hunting is very popular, but controlled. Only certain numbers of deer can be harvested and only at certain times during the year. Deer populations are very healthy, and the deer are healthy.
The opposite situation exists in part of Virginia near DC where a good friend of mine lives. The animal rights activists have gotten hunting all but banned in their area. The result? Whitetail deer populations so out of control that there is intense competition for food. Deer run over by vehicles regularly, deer overrunning neighborhoods and urban areas, and he's found deer carcasses of nearly full grown deer that only reached half of their normal size due to malnourishment, before finally dying of starvation.
Which is a more inhumane situation?
I personally believe in conservationism, and hunting (th only form of population control we have for certain species) is an integral part of that. I never advocate excessive hunting or killing for sport, and only kill what I plan on eating. Sport hunting is OK as long as the meat is donated if the hunter doesn't plan on consuming it (I have family in GA who absolutely
live for deer season, yet hate venison... but every ounce of meat they have processed is donated to a local kitchen that feeds the homeless.... so hunters aren't the boogymen after all

).