What was it that possessed figures such as Charlemagne to think that their empire being divided into tiny portions would somehow be a good idea?
This had to do with their succesion law. In the Frankish laws it was set up so that all children would inherit equal parts. This started with the death of Clovis (d. 511) were his empire was devided between his sons. There would often be war between those sons and grandsons so sometimes the Frankish lands would be united under one ruler. This continued for centuries even after the Carolingians had deposed the Merovingian dynasty.
When Charlemagne was getting old his three sons had already devided their future lands but unfortunately for them only the youngest son outlived Charlemagne resulting in the entire empire being inherited by this son, Louis the Pious (d. 840). But with Louis' sons the proces started again and the empire was once again devided between these sons.
Later in the 10th century when other royal families like the Ottonians and Robertines had succeeded the Karolingians in West and East Francia did they change the succesion law into primogeniture. This meant that the land would be primarily given to one son, most often the oldest one.