From the info so far, many people of the military faction were pushing for a renowed general like Yu-hon to become the next king and continue expanding/maintaining Kouka's might.
Perhaps King Joo-nam desired a more peaceful change in Kouka's politics, or perhaps he thought Yu-hon was too ruthless to be king; and instead chose the gentler Il to hopefully lead Kouka down a more peaceful path.
Either way it didn't sit well with the militarists, who probably decided Soo-won, as Yu-hon's son, had the greater strength to lead Kouka to glory (not helped by Il's mismanagement and Yu-hon's purported death at his hands).