A good point, another good point, victim lets mugger go and mugger comes back with friends and/or weapons, no thanks. Take him out right there and then and no worries in my view.
I don't know what video you watched, but the one I saw showed him prevailing in the end, but taking the worst of it several times - after he went after the fleeing mugger. If he had lost one of those exchanges, it would not have been 'no worries', eh?
According to the news, the mugger did escape, but was later captured, and sentenced to 4-10 years in prison. We do not know if he would have been captured if the victim had simply stopped pursuing him, but by placing his life at additional risk, we do know that he did not succeed in holding him for police - so he incurred additional risk and got no reward for it.
From the strict point of view of the laws on self-defense, in most jurisdictions, the right to defend oneself ends when the danger ends. Let's play a game and pretend that when they tumbled down the stairs together, the mugger had been seriously injured. I doubt the police would have prosecuted, but I don't have any trouble believing that the mugger would have sued. Without the civil liability shield of 'stand your ground' self defense laws (if Georgia has one), he could well sue and might very well win.
As to the mugger coming back with friends - one might as well suppose that pursuing the mugger would have been to follow him right into those friends who were waiting for him in a car to make good his escape, etc. If we're going to propose outcomes based on no evidence, that is as good as yours.
I applaud the kid for fighting back. But he kept his property and the mugger fled. He won. Call 911 and get the cops. MARTA has emergency phones on every platform, supposedly manned 24/7 by emergency responders. I simply believe that pursuing the mugger was foolhardy, but brave. I'm glad the outcome was positive.