A good horse will spot and avoid holes as well as ant beds, low boggy places, and so forth. They're subject to all sorts of ailments (never had one get an infection from a stone bruise though) but they never throw a rod and they don't run out of . . . grass.
That's the difference between a living creature with a brain and a lump of iron.
I just prefer the predictability of the bike. Not disparaging horses (being here in TX, we do have a fair number of them around). As to low, boggy places - those are the most fun!