It seems logical that they would to reduce the likelihood of being hit by arrows as well as make it more difficult for direct assault from ladders. The best reason I can conceive that they're never portrayed in movies or games or present in the remaining ruins of castles is that 1000 year old timber probably has long since rotten away before the stone began crumbling underneath it.
But at the same time you'd also expect holes for beam support to be easily added or removed on the feet of the battlements. So I guess my question boils down to two things:
Did old castle battlements have temporary overhead protection in the form of timber or ceramic rooftops?
If they didn't, would the reason point to impermanent architectural designs or would there be an actual strategic disadvantage from using them?
What you are describing is a 'hoarding', these were temporary wooden structures installed on walls when necessary to give defenders additional protection as well as providing murders holes where you could drop rocks; hot oil or sand; or arrows down on attackers from above. If you look at some naked battlements you can still often see the holes on the wall where supporting beams for hoardings would be installed.
