Regardless of whether they're really birds or they're people, page 5 has a family chart that shows that there is another heir apart from the one that died in this chapter i.e. the dead king's half brother. So there really isn't a problem with the monarchy of the country.
If they really are birds though it probably just means they're the last of their truebreed. The egg from the surrogate mother is probably not a truebreed and so is the half-brother since he also was from a concubine.