Consider cows. We eat a lot of cows. In a few years, we would make cows go extinct. Except we also breed a lot of cows so that it doesn't happen.
While many animals are far harder to raise than cows, the difficulty results in lower numbers and thus increased rarity which thus increases the price of the animal once brought to maturity. Any animal that is risking extinction from being hunted should be sufficiently desirable to make this equation balance out to a sufficient population of the animal being raised.
The only issue I really see is that in the long term this will likely lead to such species being slowly domesticated.
While many animals are far harder to raise than cows, the difficulty results in lower numbers and thus increased rarity which thus increases the price of the animal once brought to maturity. Any animal that is risking extinction from being hunted should be sufficiently desirable to make this equation balance out to a sufficient population of the animal being raised.
The only issue I really see is that in the long term this will likely lead to such species being slowly domesticated.