The sprinting mechanic you're referring to (where it usually becomes harder to shoot while doing so) exists because it adds both an element of realism and strategy. You have to decide when moving from one place to the next, do you want to move as fast as possible, or do you want to be able to defend yourself as soon as possible should you come under fire?
Your hatred of the mechanic is a preference that those who prefer more arcade-y style shooters have.
Try running from cover to cover in an online multiplayer game like Black Ops. A human opponent can blast you away EASILY... but since the programers of a game make sure the AI can't hit you when you are running from cover to cover in SP. Hell, have you ever played a game called Battlefield 2? If you got up from a sandbag cover, you could instantly be killed if someone had a beat on you. It's more realistic to lob a grenade or just shoot the person from your current "cover".
But if you prefer shooters that have sprint as a feature and you like it, great. I know that some games work with a sprint feature. Crysis is a good example because you can really fast up to a lone enemy and punch him in the face! Cover to cover sprinting just looks and feels retarded... and it's retarded that AI enemies can't hit even though they KNOW you are there.
I'm confused about your point. You jump between how the sprint mechanic works in single player and multi-player like the two are commensurable.
Since nobody cares about single player in an FPS game, I'll just talk about multi-player and not bother talking about how AI works in games.
If you're trying to argue that an enemy isn't harder to hit when they move faster....you're just wrong. I'm not going to bother explaining why, because it should be obvious.
Your example of "If somebody has a beat on you, sprinting from cover to cover is bad" is also besides the point, because that isn't a situation you'd be moving cover to cover from to begin with. That's like saying, "The sniper rifle is crap because if you're up against an opponent using a shotgun in close range, he will destroy you."
In any case you didn't refute my points that it adds an element of realism (because you can't, people in real life have the ability to sprint, and doing so they can't fire their gun with perfect accuracy), nor strategy (since both your examples are situations in which sprinting isn't what you'd want to do anyways).
All you've said is that you have never encountered a sprinting system implemented in a way you like.