Officers in the 18th century European armies didn't wear facial hair, because they were gentlemen, and it was not fashionable for gentlemen to wear moustaches or beards, from the early part of the century up until the very last years, when fashions changed as a byproduct of the French Revolution.
The only exception was Hungarian officers, and hussar officers in other armies, copying the Hungarian fashion.
In the armies of the German states, as well as the Austrian army, grenadiers were especially encouraged to wear them, and often a unit would supply its rank and file with false moustaches, to maintain a uniform appearance. It was thought that a moustache enhanced a warlike appearance. The hatmen or musketiers were also encouraged to grow them. That also applied to non-commissioned officers.
The Brits were an exception, they tended not to have anyone grow facial hair at that time, but then, they followed their own fashion, separate from the Continent.
In K&C's Hessian series, the mounted officer and the officer carrying a spontoon are both incorrect on this detail. The ensign is correct, as is the officer with a field glass.
The officer with a spontoon is also incorrect, in that he's wearing boots instead of shoes and gaiters. Only officers who were routinely mounted for duty would wear boots, typically from major on up, and sometimes a captain. Otherwise, captains and lieutenants wore shows and gaiters.
Same goes for enlisted personnel, except for Jaegers in some of the armies. Some wore boots, even though they served on foot, because they could be mounted as necessary.
Prost!
Brad