Even over 50-60 years ago, we used to, every time we rode, winter or summer, wrapped legs or not, school horse or race stable, immediately run several minutes of cold water on the legs.
That was like cleaning feet out, every horse we worked with had feet cleaned out before and after being handled and legs cooled with running cold water before being put away.
We used to do that to cool down and “help keep their legs tight”.
I have not seen that be standard practice in the US, more occasional one, other than maybe at the track.
Interesting that shows that was a good basic practice.