The point of a cooler is to keep your horse warm during a ride, ie, at a show when your horse is already warmed up and you are standing around wait for your class to start. So your horse's muscles don't cool off and you don't have to warm them up again.
If your horse is sweaty after a ride what I usually do is put an Irish knit on them and then a cooler on top. It draws all the moisture off the horse and sops up in the cooler.
I wouldn't turn out in a cooler just because it will most likely get ruined.