Can't say I'm a fan, given the potential effects on the flight-deck environment and staff morale.
These folk can afford their own Citation Mustang or Phenom or whatever, why can't they buy one of those and jet themselves around Europe for fun and kicks if their Being The Monarch day-job isn't sufficiently entertaining or fulfilling?
Perhaps the King could even offer a Part-Time Monarch job to a 'normal' to cover for him whilst he's away having fun. Surely he couldn't object to someone else flying the country for him.