"There are leg covers specifically designed for motorbikes" Yes there are such devices - they are normally called waterproof overtrousers.

I have seen a more elaborate, non-trouser solution (i.e. fixed to the bike, I think to the crashbars) on actual motorbikes on the streets of Istanbul in winter, but unfortunately I dont think I got any pictures, and cant remember exactly how it was arranged, but did think it might be useful in the UK.

Rather unlike these Turkish tarpaulin things, scooter skirts in Taiwan are mostly so girls can stop their legs getting tanned, which is conventionally considered unattractive in Tchiwanese culture, being associated with field working peasantry.