To be more precise, the idiom is typically “ride roughshod over” rather than “run ...”, and roughshod is typically written as one word.
Roughshod means the horseshoes have their nails sticking out the bottom to help prevent slipping, so you can imagine trampling someone with those could be painful.
Horses are heavy. Being trampled by one is going to be injurious or lethal regardless of whether or how it is shod. (Most horses will go far out of their way to avoid trampling a human, though; cavalry horses had to be carefully trained into it. Treading deliberately on one's foot is another matter, but, like some humans, some horses are just assholes.)
The idiom refers more to what a roughshod horse will do to a road or trail surface; the nailheads dig in and scatter surface material every which way, leaving behind a hell of a mess that'll turn to deep slush or sticky mud, depending on the temperature, with the next precipitation.
Roughshod means the horseshoes have their nails sticking out the bottom to help prevent slipping, so you can imagine trampling someone with those could be painful.