Maybe someone who has attended a War College might comment, but my impression is that, like Forrest, you do pursue a retreating enemy. Two opposite examples that I can think of is the failure to follow the Union Army to Washington after 1st Manassas, and the pursuit of Bragg and the AoT after the Battle of Missionary Ridge.

Maybe it is the doctrine that an attacking army is the one to pursue a defeated enemy, while a defending army doesn't pursue when the attacker retreats. Maybe that is something we are missing. I can see this being a general doctrine, in one way, because of the threat of ambush. That a retreating army could lead the hunter into a trap. But as you have pointed out most of the excuses given do deal with the fatigue of the defending armys troops.

It could well be when you tangle with a wildcat, you are just as happy to see the wildcat go and leave you alone.