I think that, while the "demons" are a thorn in his side, they're much more a pretext than a true obsession.
He plots to take the power from his father and uses his father's obsession with the elves to erode his power and gain the trust of his subjects
When his sister tries to win the respect of their father by going on a mission to exterminate them, he uses her project to get rid of her and become a hero to his father's subjects.
When his plot is uncovered by his father, he kill him but hides his murder under the guise of the "evil"influence of the "demon's Sword to gain the respect of his subjects.
His expedition against Ember's tribe is devised to establish his authority and prove himself to his subjects.
Then, in Mender's tale, his sons and him pretend to go on a joined campaign to eradicate the elves, but it is actually to draw away the army of the town they want to conquer and then, turn around and slaughter its soldiers and its chief and destroy the town.
The elves are a afterthought. Sure, he must eventually kill them in order to prove to his people what a brave Djun he is, but it isn't exactly his purpose in life. What he wants above all is absolute power.