Just so. Here are a few more thoughts that might help Frederic.
The Prussians made the decisions that they wanted the frontage of a double zug column and that they wanted to be able to ploy and deploy on the centre, hence the angriffekolonne which was identical in structure a French colonne d'attaque for an 8 company battalion. In the angriffekolonne, therefore it was 2 zugen wide by 4 zugen deep. Each zugen was in two ranks since the third ranks were detached to the flanks and formed up in separate sections for use as skirmishers. Even if the skirmishers reformed, I understand that they first did so as their sections (one per zug) and that these sections then fell in on the right and left flanks of the division in which their zug marched.
So, a battalion which was still in three ranks could form an open square from column at full zug distance by wheeling up the zugen from the two middle divisions and closing up the rear division, providing a square with two zugen on all faces in three ranks. That would be fine, except the battalions did not fight in three ranks. They fought in two ranks, with the third detached for use as skirmishers. This means that a square formed from angriffeskolonne at full zug interval would only be in two ranks. That was not considered very safe. Because of this, the Prussians dtopped uing the open square
Therefore the Prussians had to find a way to create more than two ranks when forming square.
The British solution was quite elegant but was on the equivalent of a single zug frontage. The Prussians wanted a double zug frontage, therefore a column at quarter distance was not acceptable.
How else to form at least three ranks on each face from the Prussian column when each zug was in two ranks? Only two ways are available. In one way, the zugen could form three ranks then th battalion could form square. That was not a good option because it required time to complete the change to three ranks and then even more time to form square (never mind it was much more complicated). Complicated methods are a bad idea in battle. The second way was to use a closed column. It should be noted that such a column did not imply all 8 ranks were closed up like a phalanx. There was some space between each division to accomodate the supernumeraries. To prepare for cavalry, the battalion only needed to halt, then outer files would turn outwards and the rear company about turn. The small intervals were filled by one of several methods: supernumeraries, files moved by supernumeraries, or skirmishers coming in. This formation had some advantages including that it could be formed even more quickly than square from column at quarter distance (which itself only needed seconds) and it did not require that precise quarter distance intervals had to be maintained. Because of this it was very suitable for less well trained units. It had a few disadvantages, including that it provided almost no firepower to the flank and that it was a very dense artillery target. The issue of being an artillery target cuts both ways though ... it was also a small target and could thus be more easily tucked into folds in the ground to take shelter.
Why did the British use a narrow front column and the Prussians a double front column? It had to do with how the infantry was intended for use on the attack. British commanders used columns for movement and waiting. In general, action was in line. The Prussians did not intend to attack in line. The 1812 instructions were that the brigade was to win the firefight by action of the fusilier battalions, augmented if needed by the skirmisher zugen of the first treffen. The non-fusilier battalions were to stay in column. In this formation they provided protection from cavalry and enabled free movement forward of Prussian cavalry and artillery. They would also be able to fire against any enemy counter attack.
Both the square formed from column at quarter distance and the closed column were useful formations. Each had some minor advantages over the other, but the driving factor had to do with the depth of the column than anything else. Quite simply, a British column on company frontage had enough elements to form open square, the Prussian angriffekolonne did not. Therefore each army used the formation that wa best suited for the columns in use.