My thought is that leashes are used for dogs as a way to show authority. It's a 'training' tool, and I don't train my children. If I see a dog on a leash tugging and pulling in the opposite direction, I chalk it up to bad dog ownership or inexperience in dog training. The leash is there to be slightly tugged at any time the dog wonders a bit from your side. You stop, dog stops. You walk, dog walks. You run, dog runs. The dog isn't even supposed to be allowed to poop on walks because you have control over him just from the use of the leash.
I own 4 dogs and have been working professionally with animals since I was old enough to legally work, so I feel as though I'm qualified to comment on this.
First, why wouldn't a dog be able to poop on a walk?! That's crazy. Isn't that the point of walking your dog, so they eliminate outside, and not on your carpet? Most responsible pet owners carry poop bags so they can clean up after their dogs. I've never met a dog owner who discouraged their dog from pooping while on a walk, unless of course they stopped in the play ground and the dog was trying to pop a squat in the sand at the bottom of the slide or something.
And second, leashes are not an authoritarian tool. LOL. Goodness. They are, first and foremost, designed to keep dogs safe. That is their number 1 use. No matter how experienced a pet owner someone is, or no matter how good they are at training their dog, some breeds HAVE to be on a leash at all times or they will be gone, dead, injured, hit by a car, etc. Scent hounds come to mind.
People train dogs to walk nicely on a leash because it aids in keeping dogs safe. Some breeds are large enough to drag an adult around by the leash, why wouldn't they train them? The purpose of a leash to keep them safe is pointless if the dog will use that leash to drag its walker into traffic. They aid in controlling dogs to keep them safe, but the way you've outlined their use as being some sort of torture tool to make a dogs life miserable is so inaccurate.
Third, the vast majority of dog owners want their dogs to have FUN and be relaxed on a walk. Expecting a dog to perform so tightly in obedience that it may not eliminate is usually reserved for competition rings, not walking in the park.
And lastly, comparing dogs and children NEVER works. My dogs also eat off the floor, lick their buts, roll in bird poop, and get their baths outside with a garden hose. Their is NOTHING remotely similar between them or my preschooler, even if some of the things I use on both might share the same name or look similar or serve similar purposes.
A leash on a dog and a leash on a child do not at all serve the same purpose. Comparing the two to vilainize child leashes is a comparison fail. Apples and oranges.