No, actually, on those occasions where the south did make it to the defensive line the North broke. It happens, it's why the bayonet charge was still a thing (and still successful) in the Falklands War, where it worked all six times it was launched. Uphill in many cases.
Ok, I agree that the South could overrun North lines and make them break, (happened many times before) but units of the Army of the Potomac usually only broke when either the North lost in the melee or they were outnumbered and (I guess you can imply if you're outnumbered) were going to lose in the melee if they didn't run before the south got there. In this case, the North should have a tactical win, losing less guys that the other side and having an edge in the resulting melee.
Of course, Goose Green shows the side that would win a melee sometimes breaks. But that's the exception, if a unit will win in the fight, they stay. Of course, they could lose by luck or missing that cavalry unit coming in while they are distracted, but if there is only one threat and no reason to believe there are other immediate threats, if a unit can win a fight they will usually stay.