Considering he's in competition with his siblings for the status of their father's successor, he actually can't rely too much on his place as just "the son of the Demon Lord". There are six other children of the Demon Lord, and only one of them can succeed their father, after all. The others will, in all likelihood, be killed, along with their supporters.
So what the MC is doing here is proving himself. Proving he has the mindset and charisma of a Demon Lord. Proving he has the ambition to take what he wants. But also proving his magnanimity, and willingness to reward those who give him what he wants. Because that's how you make loyal supporters and allies.
Agreed. If he had just walked out of the place with just his status as the prince preventing anybody from stopping him, then he would have indeed have just played the "my father" card.
But he didn't. In fact, he could have, but he actually turned around and gave Shidar a carrot. Now Shidar is basically his man...and notably, not his father's. Because Jibalgias is the one offering the Metaphase on his own. He's not saying he'll talk to his father about it, this is a deal between only him and Shidar.
Last chapter I noted that this went a bit differently in the WN. Now that there's nothing left to spoil, I can point out how, and its relevant to this. In the WN, Shidar had a much stronger spine. He continued to demand that there has to be some recompense for losing the saintess, even when Jibalgias had already pointed out why they would never be able to make use of her.
In the manga, Shidar certainly seems more like he's ending up as a lackey who lucked out when Jibaligas decided to make use of him when he said he would be the representative. In the WN, the dynamic seemed a bit closer to equal to me, or at least that Shidar was a better politician than in the manga.