Potentially interested, though I share a lot of the same questions Xalt has. In addition:

The fact that the royal line are the ones who have the power of the Runes helps a bit with the idea that the nobility specifically want someone of that same bloodline to be found, since I imagine that power is either very useful or outright required in some manner---as a deterrent against enemies, a means of controlling the knight commanders who were given Lesser Runes, etc---but is there any particular reason the nobles are actually cooperating to find the lost heir, rather than vying for the throne themselves?

Unless the power of the rune somehow makes it so that each king can only have one child or something, or the inheritance is specifically determined BY the Rune's presence and thus other royal children aren't actually considered royalty, there surely would've been other members of the royal family who branched off, right? King I has 2 kids, one becomes King II and the other becomes Duke I, but if King II dies before having a kid then Duke comes back to become King III, and so on. That's how "noble families" become nobility in the first place, typically, so is there a reason that the Dukes, Earls, Counts, etc who have closer ties to the royal line aren't waging a civil war to put themselves in power?