In addition to just death rates there are broken bones as people fall off (possibly less of a problem with buggies as mentioned above), all kinds non-fatal injuries because of biting, kicking or just panicking and either running over someone or dragging them through something without necessarily killing them.

Ulike cows and sheep who only have front teeth in their lower jaws horses have teeth on both sides and can easily take a bite of someones arm if they snap.

Add to this the massive hygienic problems that would occur if we suddenly had these animals walking around leaving their droppings everywhere.