Traditionally Grand and Archdukes were made to create a line of succession or a steward outside the royal line. A quick glance at a wiki entry states that the title goes back to the Carolignian Empire, and was an honorary given to the lord of an extra large piece of territory within the empire.
To me, and this is just my interpretation so take that for what it's worth, I think the concept was to create a ranking out side the royal family that had political power to rule. As per your post, it's why princes and princesses have, were, or are given other titles assigned to territories (both in the real world and within CT's Imperium).
I also think the idea was to prevent a power grab by any of the monarch's offspring who possibly was unfit to be king / high king / emperor, as well as create a steward who could manage the realm until replacement for a deceased monarch could be put upon the throne.
For the Third Imperium it seems like if Strepheron or any of the other emperors had a son or daughter, and if they were mercenary enough, that they could have travelled space throwing their weight around without a grand or archduke to keep them in check.