From my understanding of big game hunting, the government of whichever country you are going to go to has to grant a license to kill a particular animal. They have to cull a certain amount of animals per year because as bizarre as this sounds, parts of Africa aren't as big as they used to be. The wildlife used to be able to migrate from one side of the continent to the other to maintain access to water. There is now a road network and development across the continent that largely prevents this. Any given area can only support a limited number of animals.
So rather than culling the animals themselves, the governments of some of the countries use this license system to supplement their income. Whilst they are doing this there will be no shortage of americans with huge bank balances that want to go to africa and pretend to be the great white hunter. Sadly I don't see them giving up this source of income. There is a solution to controlling the number of animals that inhabit an area, as this is often done with elephants, but can largely be done with most known big game in an area. The females can be darted with contraceptive drugs. In the case of elephants this was done yearly. By controlling the birth rate in this way the number of animals stays known. However the number of permits sold drops to zero.