For one it realy is something that depends on the global and local region. There are multiple studies that point to a lack of evidence towards a clear answer. I’m not invested enoth to hunt down to many examples, so I’ll just quote this 2016 Australien study:
Public lands in Australia are increasingly being made available to recreational hunters to take introduced mammals such as wild pigs, goats, deer and canids. These species can cause substantial damage to environmental or agricultural assets, and it has often been argued that recreational hunting contributes to the amelioration of these impacts by reducing pest population densities. This position has been vigorously disputed by some parties. However, there is little locally-relevant evidence to support either side of the debate, and hence little evidence on which to base useful policy.
Even clearly pro hunting websites have liste of pros and cons to hunting as pest control, like this one