The question: 

Do our loglines need to be about the story's protagonist?

Here are the two I'm considering using: In Imperium, the kings have always held the Gods' favour-and the citizens' too. Yet a civil war is brewing and everyone will have to be at their best to survive, much less win.

OR

Avery Grace, the daughter of a Duke, has to learn to fight for her family and close ones, all the while coming to terms with the fact that she's betrothed to the Crown Prince-

Feedback: 

The logline does not HAVE to be about the protagonist, it just needs to reflect the plot as accurately as possible. The more POV you have, the more difficult it becomes to put the/a protagonist in there. 

From your two suggested loglines I like the second MUCH better, because it reflects personal stakes. You could choose your first, it's not wrong. But I find it vague and since so many people are involved, the stakes are too low. If you find a way of raising them it might work, but I prefer the second one

However, again we have an issue with the stakes. Plus, some edits are required. I would start with 

Avery Grace, the daughter of a Duke, has to fight for her family and loved ones... Here, you might want to be clearer as to what "fight" actually means.  Does she have to pick up a sword? Throw soup plates? Hand grenades? It's important to generate images. Images create hooks. 

"all the while coming to terms with the fact that she's betrothed to the Crown Prince"

The part in parantheses doesn't describe stakes. What's wrong about being betrothed to the Crown Prince? If he, however, is somebody she hates, or the betrothal is against the will of the Gods and is most likely to destroy the country, THEN you have stakes. 

So, I suggest to rework the second part to say 

"while getting to grips with an unruly fiancé, angry Gods and and .." I made up the last part, you can add the real elements from your plot. But I hope that gives you an idea of how to adjust. :-)