To an American, the idea of an elected monarch is pretty strange. Dare I say foreign? An inherited throne (or chiefdom, etc), is pretty common across cultures, but it’s certainly not the only way someone can come to power. Indeed, the idea of primogeniture (lands/titles passing to the oldest son, possible the oldest daughter) is pretty recent in Europe. Before this practice, a man would often divide his lands among his sons, which lead to a lot of infighting (this is what happened to Charlemagne’s empire in the 800s CE).
The 800s was a time of great change for Europe. Alfred the Great united the Anglo-Saxon kingdoms in England; finally the different tribes and groups were united under one ruler. For the next two hundred years, the nobles came together to elect their king. (Except for a brief period in the early 1000s when the Danes ruled Britain.) Much like we see in Orzammar, the king could make a note that So-and-So would be a worthy successor (sometimes this was a son and sometimes not), but ultimately, it was up to the nobles (at least, a claimant needed their approval). In England, this group was known as the Witan, who acted as advisers to the king.
That said, the nobles did often like to keep the royal line confined to one house — as we see within the Landsmeet, some nobles want Alistair on the throne because of his blood, not because he has any particular qualifications.
In England, in 1066, a similar crisis arose. Edward the Confessor died and it was unclear who he wanted to be his successor. This led to infighting among the possible claimants. The Witan did approve the succession of Harold Godwinson, but this did not end the fighting. One claimant was a French noble, William of Normandy, who invaded and eventually took control of England, ousting the native English nobles.
This kind of battle and infighting is what both Loghain and Arl Eamon fear. They are on opposite ends of the same spectrum: Loghain thinks he and his daughter should rule — Loghain, of course, helped drive out the Orlesians, and Anora has been ruling in some capacity for a while. Eamon sees the monarchy as being so new and flimsy that, in order to continue, a Theirin should be on the throne. Continuity can help hold the people together and keep out foreign invasions.
As the Warden, the player can see larger issues at work (Darkspawn!), but history shows us these were very real concerns for medieval people. Succession wasn’t simple, and the wrong ruler could spell disaster.
A scene from the Bayeaux Tapestry, which depicted the important events of 1066. From Wikipedia.