Draupne is right, that "white" horses are actually albinos, and they have pink
skin. I've had several grey horses in my lifetime (I have a lovely 17.1
hand rose grey trakehner gelding now, in fact--bigger by several inches
than the horse playing Shadowfax!), and they do turn either flea-bitten or
white-looking, though the underlying skin is usually NOT uniformly pink.
Albinos, I think, might be a "color breed" (same as palamino, pinto, etc.), but
most breeds of horses can, in fact, be varying shades of grey/turning white or
white-looking.
Interestingly, the Lipizans are born black, turn grey, then almost white as
they age. But they do LOOK white, which is probably how Shadowfax (and
Snowmane) looked.
Both the "rose grey" horses I've had were born looking like chestnuts (sorrel,
reddish), then gradually turned almost white. Fascinating to see pix of
them; they look like different
horses.
Not all those who wander are lost.