The claim here is that hunted meat is ethical because:
Almost nobody hunts and otherwise eats completely vegan at restaurants, friends' houses, etc. on ethical grounds. That person just doesn't exist. So this isn't then a great argument to use because it defends 1) the vast minority of meat consumed in developed countries and 2) an even smaller minority of people's actual positions.
Hunters aren't picking out weak animals that are terminally ill. Generally, the most desirable animals are the most virile, strongest, and largest. So this isn't an overpopulation control or act of mercy on the animals, so long as people brag about how many points that buck had. So any argument about controlling weak animals is even less likely than the argument as a whole.
The number of deer that are hunted in the US is about 6 million, which makes 336M lbs of meat, or just about 1 lb per person (This is much larger than other animals like elk for instance, which is about 6M lbs of meat by my calculation - assuming 400lbs meat/elk - in states when it's common). At the same time, The US consumed about 265 lbs/person of meat. So if everyone just relied on deer hunting, we're looking at a 99.5% reduction in meat consumption in the US by these numbers.
So again, the plan that carnists propose to avoid going 100% vegan is to go 99.5% vegan on meat (and 100% on other animal products). So even if none of the other problems with the argument existed, it's a defense for such a teeny tiny minority of the animal products that are actually consumed.
We don't consider that painless or quick death is much better ethically than allowing one freedom, even if that freedom comes with risks that one might die. So this argument just doesn't work in any other context. For instance, no one considers shooting stray dogs or cats to be ethical because they might run into some other trouble in the wild, and then turns around and uses this as a defense for dog fighting. And no one would say "we shouldn't release wrongfully imprisoned people because they might go out there and get in car accidents or mugged". It just doesn't work for literally any context other than consuming animal products.
This defense can be used to justify, for instance, stealing a car. That person was going to lose possession of that car at some point anyway, whether by the car breaking down, the person dying, or whatever (if you argue they were going to sell the car, well whatever they spent that money on would reach the same fate). Also, everybody dies; so, now murder is okay! Again, with this reasoning we can justify literally anything.
Hunting and hunting-related practices such as the sale and import of deer carcasses have led to an exacerbation of chronic wasting disease. Attempts have been made to control this by banning certain practices around hunting like baiting. So this still hurts the ecology.