Ah, okay, right. I think that explanation does make sense in-world, but I think my fanon explanation for it is that as well as being human (Sons & Daughters) there's also a blood-right aspect, the same as any monarchy, and that when Frank & Helen's line fails, there's no one in Archenland who fits the four thrones requirement--in other words, possible intermarriage with non-humans doesn't obviate the Sons & Daughters thing. The Calormenes are excluded in TLB since they don't have that blood-right of Caspian's line, but Caspian can reign because he is a Son and because he's been bestowed the blood-right by Aslan, under Peter.

I mean, we're basically on completely fanon ground here, but I think the Witch probably killed the Tree and then went after the realm. Basically, cofax7's The Last Defense of Cair Paravel is my default assumptions about how the Winter started.