This is here so I can point to it if/when people have questions about my reasoning. I claim no revolutionary new ideas. In fact, this has probably been hashed out by others. :D
1: Begnion, ruled by matrilineal descent
The definition is here, and I don't think there's much to argue about on this point. The implication in the timeline is that the Apostles are all female (the Japanese seems not to be gender-specific, so I guess this could be worked with?), and we know every ruler since the establishment of the empire was from that line.
The point: it doesn't matter who an Apostle might marry once Daein and Crimea pop up. Not that Begnion seems to acknowledge them enough to consider political marriage, but if they did, I see lesser children (i.e. not Apostles) being pawned off for such arrangements. The Apostle's only familial duty would be to provide her successor, and since the ability to hear Ashera is passed on through the female half of the line, the male part frankly wouldn't matter as much.
So hey, I wonder what would happen if all the female members of the family were killed off but there were a few male relatives left to carry on the line? Maybe I'm being too specific here - maybe it's just the first-born of a generation in general that manifests the talent. I don't actually recall anything that says the talent only passes to first-born.
Maybe Sanaki has some male relatives hanging around somewhere.
2: The senate vs. the throne
There's a story on FF.Net asking why it seems two kingdoms were founded. (Haven't read it yet, but I will! Eventually.) I think the simplest answer is best, and it's the one I keep in mind - Altina founded the country, but later the senate transformed it into a theocracy and sat the Apostle on the throne. All power is given to the senate in that entry of the timeline. The Apostle seems to have ceremonial and popular significance, but all practical/effective power seems to go to her so-called subordinates. It doesn't seem to have ever been different.
Sephiran was probably trying to change that and failing. It's notable that Elincia doesn't actually get much help from Begnion until he returns and smacks the senate around, and Sanaki clearly has to work around her own government to get anything useful done. Her authority only extends as far as her personal presence, and she was probably always one step away from a knife in the back. It's interesting Sephiran's existence alone was enough to protect her from that, and they didn't try to off her while he was gone. I'd speculate he was in league with the senate, except that a) Lekain had no clue what was going on in the Tower, and b) Sephiran seemed genuinely interested in toppling the them in her favor. It wasn't just something he did to cause chaos, which was my original assumption.
How Sephiran managed to gain power in canon is a big frigging mystery to me. There's no logical explanation for it.
Why isn't Sanaki more paranoid? She really should be.