[A_Brandybuck]

The origin of Gil-galad may be a huge problem, because Tolkien made it in a very confusing way and Christopher took up an unsatisfying solution in the Silmarillion.

But I don't think, that this is the problem of the question "Who's King now" after Turgon has died. When Turgon died, Gil-galad was the only male descendant of Finwe in Beleriand besides the sons of Feanor, who had given up the title after Feanors death.
Fingolfin's line was dead: Fingolfin, Fingon and Turgon were no more.
Finarfin's line was also dead: Felagund, Orodreth, Angrod and Aegnor were no more.
That made Gil-galad automatically King of the Noldor in Beleriand. Earendil was too young and Elrond was born later.

That leads me to the question, could only Male claim the Kingship of the Noldor? Because there were also Galadriel, who is the daughter of Finarfin. But it seems to me, that she could not claim a kingship. Look, Galadriel and Aredhel were the only Women under the descendants of Finwe (not considering some other versions in HoME), but they couldn't claim their own kingdom in Beleriand or rather the Lands of the North like the male princes. Galadriel was somehow "attached" to her brothers and later to Doriath and Aredhel was "attached" to her brother Turgon.

When they are not allowed to or couldn't reign a realm, why could they claim the High Kingship of the Noldor in Beleriand?
Consequently, there was also Gil-galad to become High King of the Noldor.

Concerning the successor of Gil-galad, I agree here with Formendacil, that Elrond refused, because there was no 'real' Kingship more.