As someone who spends my working life with my arm up a cows backside I would beg to differ on this. Horses (and cows) consume vast amounts of fibre rich grass/hay/sileage.....ie vegetable matter and it's in the mouth and out the other end within a couple of hours. Dogs and cats on the other hand consume meat. Their excretea is far more unpleasant to deal with and as someone else pointed out can also contain harmful toxins/bacteria.
As a child, the entrance to my junior school ran alongside a getty, this ran along the backs of rows of terraced houses. The residents put their dogs out into the getty to do their business, the smell in the summer with the hot sun beating down made the getty not a nice place to walk. Horse (and cow) poo is relatively pleasant in comparison............well I know which I'd rather have my hand in
Frisbee