Having discussed this with @warsofasoiaf and doing a bunch of research, my conclusion is that the three battles formation isn’t an issue of tactical effectiveness as much as an organizational issue, whereas something like the flying wedge (which maximized the effectiveness of a knightly charge in forcing a breakthrough in the enemy line) or a shield-wall (which was both an excellent defensive tactic especially for armies relying on less experienced and less well-equipped forces) were tactical in nature.
The advantage of the three “battles” is that it’s a very simple organizational structure, which makes it well-suited to armies made up of feudal levies with relatively little experience of fighting together compared to standing armies. More sophisticated organizational forms require a good deal of coordination, which means you need an officer corps to make sure that all the different parts function. The Roman legion’s famed flexibility on the field, for example, would never have been possible without an elaborate officer corps of optios, centurions, military tribunes, etc. who were given wide discretion in the field to ensure that these kind of complex formations worked:
By contrast, a three-battle system didn’t need that level of complex organization – you had your overall general, your two or three commanders, and the men under them. The three battles had set locations (the vanguard at the front of the column or at the right of the line, the middle in the middle, and the rear-guard at the end of the column and on the far left) and set responsibilities that everyone knew ahead of time (the vanguard was responsible for scouting and foraging, the rearguard covered retreats, etc.) which meant that you didn’t need very good communication and coordination, and they were also political prizes that generals could dole out to reward their vassals (the vanguard being the place of honor).
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