Just the similarity between one group of arrogant fools, and another.

We can discuss what is "sporting", but that's a human concept and I've never heard animals discussing it. Does anyone really think that the animal predators worry about it? All they worry about is whether the next meal is there, and an easy meal beats a hard one every time.

"Fair Chase" is a concept that has some value.
1. When game is not unlimited, it provides a mechanism to reduce over-exploitation. Game is a public resource so there is an argument that it should be shared, not just grabbed in bulk by the first people who can - by any means.
2. It adds value. Hunting as we do it is not about meat, but about memories. Fair chase as a principle points us towards putting in effort and earning the animal, which creates better memories. What we work for, we value.
3. If you regard trophy hunting as some kind of dick-measuring exercise, instead of an indictator of personal effort, then you may want to ensure that the man against whom you are measuring yourself has played by some form of "rules".

Cougar have plenty of opportunity to get away and the hunting - particularly dry-ground hunting, is notoriously hard. It's sporting.