http://www.uesp.net/wiki/Lore:Amulet_of_Kings "The amulet may only be worn by certain individuals- those who can be said to have the 'Dragon Blood' in their veins. It is a popular misconception that this refers to the heirs of the Septim lineage. This is obviously false since the majority of wearers were not Septims."
The short answer, I am sure other will provide more detail.
Either that or the Arcturian Heresy was wrong. It could happen.
Either way, the Amulet was probably so that as long as Cyrodiil had a ruler, and therefore order, the Divine barriers would keep Mehrunes Dagon from invading. No point in shielding a land when it would be an improvement to get taken over by Dagon. Of course, the Mythic Dawn sort of found a loophole; assassinating off those that can wear it, and when Akatosh was summoned/mantled/whatever, it was a situation where the Divine Barriers no longer worked as they were meant to, so when they were re-established, they would supposedly never be destroyed.