To begin with, it is important to understand that the SCA, god bless them, are not really an effective means of gauging how medieval battles were fought. They are fighting with blunt weapons that, although they can leave a decent-sized bruise, are not really lethal. As such, participants are much more eager to approach an opponent since the outcome does not involve them lying in the mud clutching their now-exposed intestines. Another is that members of the SCA can afford a complete covering of armour, whereas such sets were rare except for a small elite throughout most of the medieval period. Lacking armour, warriors would reluctant to risk themselves in such exposed combat displayed in the video.

Now, having covered that, in response to your question, the answer is "it varied".

In battle all warriors who were not skirmishers fought in close formation. There were several advantages to this. It made a collection of men easier to command and move. It enhanced defence as shields could overlap, and it made men feel safer as they were less likely to be surrounded compared to fighting in an open group.

There were various ways a formation could be broken. Enough casualties could be inflicted that the remaining warriors panicked and fled. The formation could be outflanked by infantry or cavalry, or assaulted by a weapon they had no adequate response to.

Another overlooked factor is morale. If the men in a formation were confident and/or well-trained, they would be less likely to flee, whereas a scraped together militia could panic and run just at the approach of heavy cavalry.

During the Third Crusade efforts by Saladin to break up the Crusader column marching with King Richard through archery and harassment failed. The men kept formation and this made any direct attack impossible since it would be easily repelled.

At the Battle of Hastings repeated Norman cavalry and infantry assaults failed to break the Saxon shield-wall. An important reason for this is that the core for the shield-wall was made up of thegns and huscarls, well-trained, well armoured and experienced soldiers whose presence encouraged other less-prosperous troops not to flee.

However, even well-trained troops had their limits. Although in the ancient period, the Battle of Carrhae serves as an excellent example of what happens when an army confronts an opponent utilizing a weapon they cannot counter. The Roman heavy infantry under Crassus had almost no long-range troops. This meant Parthian archers could continually barrage them with missiles with almost no risk. The Romans had to remain in close formation to protect themselves, but as a stationary target this made them more vulnerable. Parthian horse archers could approach close to them and loose a few arrows. At close range these arrows could frequently penetrate shields and armour. It is a testament to the Romans that they never actually broke and scattered, but they reached a point where they just had enough and the troops pressured Crassus to negotiate a surrender.

Sources:

The Complete Roman Army, by Adrian Goldworthy

Warfare and Society in the Barbarian West 450-900, by Guy Halsall

The Development of the Komnenian Army 1081-1180, by John W. Birkenmeier