Lots of information and ideas here to consider when evaluating ethics of shooting game animals at any distance.
Although I am fond of varmint shooting, I have never shot a game animal over 350 yds, mainly because of where and how I hunt. Many of the people who make these long shots seem to want to shoot an animal at long range because they have a rifle that is capable of very good target accuracy at that range. As pointed out in another post, targets don't move, and a lot can happen to the bullet in 1,000 yds. Wind drift, loss of velocity are considerations. And the movement of the animal itself will change the outcome of the shot. At 5 mph, leisurely walking pace for deer or elk, they will cover 7.5' in one second, or about 10' by the time the bullet arrives from 1000 yds away. That can turn a good shot into a miss, or worse, a gutshot animal. Now you have to go looking for it. In mountainous country, it can take an hour or more to cover 1,000 yds, depending on terrain.
I am not sure, but how many of these long-range shooters are careful that there are no other hunters between them and their target? If you are 1,000 yds away, and I am only 200 yds away from the animal, who is in the wrong if someone is hurt? That one is pretty clear.
I remember reading articles when I was younger that called people who shot from more than 300 yds unethical and irresponsible, primarily due to the lack of experience of the typical hunter, the equipment available at that time, and the bias of the writer. Now we are involved in the same finger-pointing. I am not sure I like pointing fingers to begin with, but judging others' actions by our experience and bias can be hard to justify in the long run.
I guess the main objection I have against shooting game at 1,000 yds is the low odds of recovering the animal if it does not drop at the shot. A wounded animal can cover a lot of ground, and shooting from 1,000 yds, you really have very little chance of seeing where the animal runs to, how they are acting after the shot, and you will likely not hear the shot strike, meaning you will not know where the animal was hit. I may frown at other people shooting from extreme range, but very little about ethics is absolute. I hunt and take shots I have the equipment and experience to be successful at most of the time, and if the 1,000-yds shooter has the experience and the equipment to make the shot almost every time, the decision is up to him.