First of all, dogs are not trying to cover their scents, or even ‘clean up their mess’, that makes no sense from a biological stand. This is in fact a very instinctive method of spreading their scent.
How can they spread their scent while not actually touching their mess? Well, they actually have little secretion glands in their paws that secrete pheromones, activated as they kick. By doing this, your dog is able to leave a scent trail behind, letting other dogs know he was there.
Each of a dog’s senses has evolved to augment its’ hunting and tracking ability (or more accurately- a wolf’s senses). While their eyesight is limited in comparison to ours, has in fact developed to pick out small, quick movements of prey, few animals on this planet can rival even your little teacup’s sense of smell. Where we see the world through our eyes, they see it through their nose.
The Canine sense of smell is so powerful, none of our advanced human electronics can match it, not one of the known great hunting cats comes anywhere close to even that little teacup I spoke of earlier. Bloodhounds are able to trail a scent as it washes up alongside riverbanks, search and rescue dogs can often follow a (both ground and air) scent from over a mile away, or even buried deep underneath snow!
The point here is, they would have absolutely no problem at all discerning where another dog has been. When wolves and dogs roamed wild, (and the ones that still do), they needed to warn other animals away from their territory. This was the dog’s’ way of protecting their food sources, and also their breeding females. You might think that the dog’s feces is sufficiently pungent to warn off competing animals, but much of the scent is lost once the feces dries out. The scent from the dog’s feet glands is more enduring.
Can you see the third Husky to the right, in the background there? I realize it may be difficult to imagine through this freeze frame photo, but he is performing that exact behavior!