It's either that or the chance of your characters actually catching it when it appears in a province. Maybe both? Unfortunately the actual details of how the game uses these values isn't shown to us.

Brittany is kind of a weird case because its a kingdom that has only one de jure duchy within it, so both the duchy and the kingdom have exactly the same borders. Do you own both the kingdom and duchy titles? How many of the actual counties do you control?

Yes, that fragmentation would be the main advantage. I wouldn't discount having the "easy" backup of the Kingdom de jure claims, though. Courtiers can die (and have an annoying habit of doing so right in the middle of the war to boot), pushing random other people's claims can introduce all sorts of obnoxious inheritance complications ("Why is the King of Bavaria about to inherit Wessex?!") and requires landing the courtiers first.