What's the purpose of adding water to an egg wash?
The proteins in a fresh egg are too viscous, even when thoroughly beaten, to wash a baked good.

The water thins the proteins so you get a nice glaze instead of a layer of scrambled egg.

Older eggs have sometimes degraded enough that they don't need the water but it is easier to always add water than to make a subjective judgement on the state of your egg proteins.