1194 Official Translation: Mihawk: "Remember this: the color of the supreme king is something so strong, the weak cannot remain standing in its presence. A sense of self."
And this is about it.
Supreme King is a result of self-acceptance.
Even if you love yourself, if you can't accept your flaws, you won't be able to use the ability.
Kaido is a suicidal drunk, but, has fully accepted that he couldn't become Joy Boy, and dedicated his life to finding the new Joy Boy. He failed with Ace, who he thought was the next Joy Boy, being Roger's son, and wanted to save in Marineford, then thought he failed again on Onigashima every time Luffy appeared to "die". This is why Kaido smiled the most when Luffy bounced back and got stronger: he wanted to see the Joy Boy he couldn't become himself.
Despite being incredibly self destructive, Kaido's CoC was incredibly powerful.
So, we know self-love WAS NOT Sanji's problem inherently. It was the solution. Sanji's problem is hating the Germa in him, same with Shanks hating his past while fighting Harald.
Kaido proves you can hate yourself/destroy yourself and still maintain a strong sense of self.
Sanji WANTS to love himself, but couldn't due to his Germa trauma. Now, he can finally accept the love of the others around him, and defend himself for them. Baby steps.
Supreme King Haki means you refuse to compromise who you are for anyone. Not Gods, not your lover, no one.
IMO, this is why we've seen Kuzan/Garp be the two Admiral level characters to display CoC.
They're the only two Admiral level Marines that will straight up disobey orders for their own sense off self/own sense of justice. They refuse to let the WG/CDs compromise who they are.