The military advises turning the map around (in the direction of travel) because it's more idiot proof, surely.
It's true that they want a method that works well for everyone, even those with poor spacial sense.
But it's also true that everyone, even people who are not idiots and are good map readers, makes errors. And while you might not invade the wrong country, it is entirely likely that in the dark, in an unfamiliar place, where there is some confusion, you might wander into a minefield or, in my own real life experience, have a driver turn left instead of right and end up in the training impact area.
If you practice walking in the woods or navigating in a car with a map that you move based on your direction, you become much quicker to notice things that are not right, errors you have made, or even changes to the landscape compared to the map. It's not just the direction of travel that is important.