Fog usually burns off before 10:00 am.
That still advancing the battle a minimum of four hours.
Even if there was ground fog, church steeples would still poke through. Ground fog can confuse ground observers. While a pilot may not be able to see the end of his own runway (while on the ground) anyone flying overhead only sees a thin mist.
The definition of a ground fog is it hugs the ground. Often you can see blue morning sky above. If visibility horizontally is 5-10 meters then the aircraft and paras can see the ground.