I will go with the others to say it depends entirely on the horse. Some can, come can’t. As another poster noted, if he is competing at a higher level he will need to be kept fit, which means clipping and blanketing appropriately will make your life a heck of a lot easier.
Personally I like snow shoes/pads on them at it seems less treacherous for them but that may depend on where you live (we get a lot of snow then mild weather which makes for wet heavy snow and therefore packed hooves a frequent problem otherwise).