I am pretty sure the song is "one horse open sleigh" , at least that is the only way that I have ever head it used. I think a dray is more like a lumber wagon, and so a dray horse would also be what you call a drought horse there, Di, and we refer to as a draft horse here. An "open"sleigh, simply means there was no top on it.
I have seen newer versions of slip on shoes, but never anything like the one pictured , and I am still puzzling how it would help the horse in either snow or a swamp. Usually, in the winter, we had the horse shoes pulled to give them better traction in the snow, and that looks like it could get downright slippery if used on ice.
Since it would be simple to take on and off, though, it could be removed if needed. The solid bottom would help the horse from balling up snow in the inside of their hooves, so I can see the usefulness there, but if they trot, they then throw the snow out. These horses were usually pulling a large load, so they probably didn't trot, so the shoes might have been necessary.
Wonderful WII, Sifu !
And I was as puzzled as everyone else. I was thinking it could be used to do something TO a horse shoe, due to the shape and size, but it never occurred to me that it was actually a clip on shoe.