I suppose it's a waste of time, but I still disagree that predators can reproduce faster than they can be done away with.
I think both sides of the argument are somewhat bogus here. History has shown that for some species, (e.g., the coyote), expensive, time consuming eradication campaigns did not reduce the population. Coyotes were, in fact, able to reproduce faster than people could kill them. But this is not true for ALL species, whether predator or prey. This is partly why we are in the midst of an extinction crisis.
There are many factors influencing the population of a species, for example: hunting pressure (by both humans and other species), quality of habitat, quantity of habitat, connectivity of habitat, road density, climate, etc. Also, the innate reproductive limitations of the species are very important: some can produce only one or a few offspring once a year. Others can have many litters of multiple young. The latter group is much less vulnerable to human hunting, because they can quickly produce more young when habitat opens up, as it does when you shoot an individual. The former group is not so lucky, and their numbers can decline sharply due to human hunting. This is why we regulate hunting.
For me personally, there are 2 reasons favoring prevention of conflict rather than destruction:
1. Many of those factors influencing animal populations are changing rapidly: climate is quickly warming, habitat is being destroyed more quickly than ever before, and human population (which impacts many of these other factors) continues to grow. We already know that many species around the world are failing to cope, and we cannot assume that species doing well now will still be in good shape tomorrow. We've managed to kill off abundant species before, and we can do it again. Take the passenger pigeon: once so numerous that people shot them out of the sky for fun, leaving many of them to rot. Current status? Gone. All of them. Combine the backyard chicken craze with climate change, urbanization, suburban sprawl, and human population growth...That's a pretty stiff challenge even for these common predators we lure in with a chicken dinner.
2. It's simply more humane, IMO, to prevent conflict with wildlife, than to kill them. This is a value thing which others may not share. I understand this is not a concern for everyone.