It seems like your problem is simply my vegetarianism. You haven't pointed to a flaw in the enactment of what I proposed: dramatically increased, state-funded sterilization efforts. I know my solution wouldn't be very agreeable to non-vegetarians, but I was hoping that people would focus on analyzing my proposed solution from a perspective of effectiveness and feasibility (even if they disagree with the vegetarian philosophy). Also, I know my proposition wouldn't go over well today, but society is dynamic and maybe someday, we will move closer to what I have in mind.
The problem is, there's really no reason for the government to take over an industry that already (A) provides a valuable boost to the economy and is well-regulated to begin with (people having to pay for licenses and, in some areas, rent booths or stalls to hunt from, for example); and (B) doesn't pose any real detriment to the environment or to the economy/society. Libertarians will tell you that it's just less cost-efficient to take people who are spending their own money hunting out of the game, and put government employees (who are presumably funded by the government) in to do the same job but for pay instead....and in a rare twist of fate, I actually agree with that position.

The hunting industry is already well-regulated with hunting seasons that coordinate with animal breeding habits to ensure a steady population, so speaking solely on behalf of law-abiding hunters, there's really no threat to the ecosystem. It's not like the chem industry with hundreds of metric tons of toxic waste being dumped into rivers and ruining entire ecosystems, so there's really no need for much more oversight than what already exists, as far as I'm aware. And it's not like healthcare or banking, where the private market's artificial tinkering with rates and gambling with profits/policies has artificially made a necessary service inaccessible or impractical to many. This is one of those situations where there's just not much of a point in dedicating government time and money to an industry overhaul.