I think Jasnah was skipped over in the line of succession only because she's already queen of Alethkar, and it would be seen as a conflict of interest for her to be in charge of both Urithiru (which is at least supposed to be independent from other nations) as well as Alethkar, even if Alethkar is still taken over by Odium. Otherwise she would definitely be the most logical choice.
But yeah, I think having something like a council of Radiants from each order would make the most sense, or one of the Bondsmiths leading (when available). From the gemstone archives, it seemed like Melishi (the Sibling bondsmith) might have been a leader in the past, but I'm not sure.
As for the Vatican comment, I could also see it working like that, where the Bondsmith's squires work to choose a new leader from among themselves, who ends up becoming the next Bondsmith and leader of Urithiru. I wonder if we'll see Dalinar or Navani gain any squires in this book that could point to that.
Really I just hope Kaladin doesn't end up having to be king. Frankly I don't think he would know what to do with that particular kind of responsibility. He was already overwhelmed with just trying to manage the Windrunners; he works best in small groups. and I just want him to get a break lol.