I fought a pretty tough battle today against rebel getains. I destroyed the getai early in the game and let a rebel 3/4 stack stay while I mopped up elsewhere. I decided to use my regional army of 2 levy phalanxes, 2 thurophorois, 1 slinger, 1 fm and 1 thrakian podromoi. The enemy army had a ton of heavy infantry and naked spearmen.

At first my phalanxes were being ripped apart by the naked spear men, I managed to kill one unit of them, but they literally fought to the last man. This also happened to a galothraikes unit They did not rout at all. Then I managed to make their general chase my fm and I killed him. During this time, our armies were resting, meaning our lines weren't engaged. As soon as their general died, the whole army of about 8 units hightailed it back to the hills. They seriously routed without being in combat.

Id like to say that not many of them left the field.

So yes, a generals death has massive impact on the battlefield. Especially if he is the FL of a dead faction.