Shaft drive has been around for ages.
One of the downsides is that it drives through 2 bevel gears, one at the front and one at the back. This reduces efficiency compared to a chain or belt drive, not sure by how much on a bicycle, but I've seen 5 to 10% quoted for motorbikes. And some shaft drive motorbikes have the engine arranged longitudinally to eliminate one of the bevel gears, this would be rather difficult on a bicycle.
As I see it, the main advantage of shaft drive over chain drive is the elimination of an exposed and messy chain. A chaincase solves that problem, like shaft drive chaincases have been around for ages. I'm quite happy with the Chainglider, a modern chaincase that will fit to most of the hub-geared bicycles I have.
AFAIK it is not possible to fit belt drive, shaft drive or a (good) chaincase to bikes with derailleur transmission.