[Finwe]

Elrond didn't really have much of a choice, because he would have been a King only in name. In Gil-galad's time, many Elves still remained in Middle-earth, but after the Last Alliance, and in the Second and Third Ages, it seemed like a mass migration of the Elves took place, and most of them left for Valinor. If Elrond had taken up the Kingship, then he wouldn't really have had a people to rule.

Accepting Elrond's right to rule over Galadriel's is a bit of a non sequitur. His High Noldorin blood came from his grandmother, Idril, not from Tuor (obviously). If Tuor had been the Elf, and the Heir of Turgon, then Elrond would have had a rightful claim to the throne. But since thrones and inheritance were reckoned through the father's line, Elrond wouldn't have been able to inherit the throne. If they tried, then Galadriel, who was the daughter of Finarfin, the brother of Fingolfin, would have had the same right to rule. In fact, she would have had a stronger claim because she was far older than Elrond, and had much more experience than him in matters of ruling.