Asgard being a monarchy, clearly someone has to, but Loki appointing himself does not make him the rightful ruler, and him lying to everyone by claiming Odin did appoint him doesn't make it so, either. Why shouldn't Frigga be regent till Odin wakes up again, for example?

It was Frigga who appointed him regent. He didn't take it for himself.

it's the shiny price that proves he's best after all, and he wants it.

I wouldn't put it quite like that; Loki did say to Thor "I only ever wanted to be your equal."
Yes, Loki is going about all this the wrong way, but it's as if he doesn't have a clue about what the "right way" would be.

(I have finally gotten around to getting some MCU icons, yay)