Forget the whole running deer vs. sitting duck thing.
The point I was trying to illustrate was that principle of takng the "surest shot" is inherently at odds with fair chase. Exactly how much those principles conflict of course depends on the situation, but I've never heard of a situation where these principles weren't at least slightly at odds.
Perhaps getting personal will help...
The one deer I've shot so far was a young buck, and evidently far from being "the sharpest knife in the drawer." He came trotting along in a harvested cornfield, coming from upwind, traveling parallel to, and 60 yards away from, the tree line I was sitting in. Everything about his demeanor shouted "I have no situational awareness whatsoever." If he'd been a human in a shopping mall, the pickpockets would have had him in seconds. I made a horrible impression of a deer call, he stopped to listen, and I shot him.
Some would say that was fair chase because even though my purpose in calling to the deer was to make him stop, it did at least let him know I was there before I shot him. Others would say I took an unfair advantage of his youth and inexperience. Speaking for myself, I have a clean conscience but kind of feel like I played him a dirty trick. The main regret I have from the hunt was that I misjudged where the "high shoulder shot" should have been placed and only hit him in the spine, requiring a follow-up shot. So even though I had a totally broadside shot on a stock-still animal, he still suffered more than he should have had to.
From my experience, the term "fair chase" is viewed differently depending on the critter you are hunting.
Generally speaking, "fair chase" when it comes to deer has nothing to do with whether the animal is standing or running, but whether he has the ability to leave the area...ie no high fencing keeping him contained where you can get at him. If a wild deer busts me, he's gone, and there won't be much I can do about it.
When it comes to birds...it's a little different. The "sporting" thing to do is shoot them in flight. That is, after all, the essence of being a wing shooter.
The only migratory birds I've shot on the ground were ones that were previously wounded.
Here are a few examples:
Fair Chase deer - The area I hunt in MS is 1,000 acres of private land with no containment system to keep deer on the property. I go to the woods with my skill set, and I hunt a deer that has his own skill set. There is no fence to keep him inside the boundaries where only I can get him. He is wild, he is not used to people and won't tolerate their presence...once he detects you...he is gone. How I kill him is up to me. Many times I won't be able to as he'll give me the slip. I shoot almost every deer while they are standing still...and that does not in any way violate the principle of fair chase...it maximizes my odds of delivering a quick, humane kill. In one or two jumps that deer can be over a ridge or in a thicket to where I'll never see him again.
Not fair chase deer - We once had a customer come in for a duck hunt. He's a wealthy dude from west Texas. He has a big "game farm" where people can come hunt exotic animals. While he was out with us he told a story about a guy that came down to bow hunt a red stag. This guy wanted to stalk and kill a red stag, which he did. The hunter came back to the lodge telling the story about how awesome it was to get so close to the animal before releasing his arrow. He spoke of how stealthy he was in executing the stalk.
Then our guest began to ridicule the hunter in the story. He said the reason he was able to get so close was because the animal was tame. They feed it by hand...it had no fear of humans, and was inside a high fence operation. That is not fair chase.
There's nothing wrong with hunting in a high fenced area, it's just not necessarily a fair chase deal. In that particular case it certainly wasn't fair chase, and in my view the hunter should have been told the animal was tame and you could walk up an pet it if you wanted to...he might have passed on hunting there if he had known all the facts.
Fair chase birds - When I hunt pheasant in SD we hunt huge parcels of rolling ground. it's not uncommon for the birds to flush 40 or 50 yards ahead of us at times. It can be tricky getting shots because they are wild birds and they do not want to be around you. They are wild birds, and shot on the fly...they have a very good chance of escaping the hunter.
Not fair chase birds - There is a place east of where I live where a guy has a bird hunting operation. You call and tell him how many birds you want to kill and he goes out and "plants" them.
The guy I work for joined this club and he relies on me for a lot of his hunting advice. I went out there before he did just to see how the operation was run.
When I got there the guy asked if I wanted to go out with him to set the birds. "Sure" I replied, and we jumped in the buggy. We drove around the property with a cage full of birds. Every now and then we'd stop at a pre-planned location and he'd take a bird out of the cage. He'd hold it with both hands, and then swing his hands around in circles real fast. Then he'd pick up a piece of wood or trash and stick the bird under it.
I asked "what are you doing?"
He replied "I'm getting em drunk!"
"What do you mean?"
He said "I get em real dizzy so they stay right here. Then when the hunter comes out they'll be here for him."
I'd never seen anything like it. Later, my buddy came out, and I walked with him as he "hunted". We got to a place where the guy planted a Hun and nothing happened. The dog was trying to flush something, but nothing was happening. Eventually the redneck that runs the place walked over there and lifted the piece of plywood and hay the bird had been placed under earlier. Out walked the bird in front of three people and a dog...it didn't fly. Eventually he kicked it to get it in the air.
It was a sad display, and not what one would call "fair chase."
These are just a few examples that might help you start seeing the "lines" if you will between fair chase, and something that isn't fair chase.
It has much more to do with whether the animal is moving or still when you pull the trigger.
I'm sure others here can provide additional examples.