After all, Tolkien wrote all of this to create a mythology for England. There are no Elves left in the world and so, Elven kingdoms no longer hold importance. Tolkien let them die out before the Elves left Middle-earth altogether.
There is evidence of this thinking on Tolkien's part...first in Volume 6 of the HoMe - The Return of the Shadow. Even though he was given Vilya by Gil-galad, Elrond never assumes that the royal lineage is passed to him as well.
An early version of a conversation between Bingo (who later morphs into Frodo) and Elrond in Rivendell...
… ‘And thither came many of the Great Men of old out of the Far West Island which was called by the Elves Númenor (but by some Avallon) [> out of the land of Westernesse (that they called Númenor)]; for Sauron had destroyed their island [>land], and they were exiles and hated him. There was a King in Beleriand of the Númenorean race and he was called Elendil, that is Elf-friend. And he made and alliance with the Elf-king of those lands, whose name is Gilgalad (Starlight), a descendant of Fëanor the renowned. I remember well their council – for it reminded me of the great days of the ancient war, so many fair princes and captains were there, yet not so many or so fair as once had been.’
‘You remember?’ said Bingo, looking astonished at Elrond. ‘Bit I thought this tale was of days very long ago,’
‘So it is,’ said Elrond laughing. ‘But my memory reaches back a long way [>to long ago]. My father was Eärendel who was born on Gondolin seven years before it fell and my mother was Elwing daughter of Lúthien daughter of King Thingol of Doriath, and I have seen many ages in the West of the World. I was at the council I speak of, for I was the minstrel and counsellor of Gilgalad…
Elrond certainly does not think himself successor to Gli-Galad, but merely his minstrel and counsellor.
We also get a hint that no Elven kingdoms are meant to pass beyond Gil-Galad from Tolkien himself, written in a letter circa 1956...Letter 183.
‘But that was not the policy or duty set out by the Council of Elrond. Only after hearing the debate and realizing the nature of the quest did Frodo accept the burden of his mission. Indeed the Elves destroyed their own polity in pursuit of a ‘humane’ duty. This did not happen merely as an unfortunate damage of War; it was known by them to be an inevitable result of victory, which would in no way be advantageous to Elves. Elrond cannot be said to have a political duty or purpose.’
So it seems to me that while Galadriel awaited her test (from Frodo and the Ring) in Lothlorien, maintaining the Elven nature of that special place while she waited, she had no designs on re-establishing any of the magnificent Elven kingdoms of the past.
Thranduil kept his small kingdom in Mirkwood, but it was not a place of splendor and glory either.
Cirdan, who once held an Elven Ring of power before giving it to Gandalf, turned from helping the Men and Elves to enter Middle-earth to helping the Elves leave.
Lastly, for his part, Elrond seems to have assumed the role of caretaker in Rivendell. We have another hint from Tolkien in that Rivendell (or Imladris), is also called the Last Homely Home. A place of beauty and rest for a tired race. Elrond serves his purpose of rearing and guarding the heir of Númenor until that day when Aragorn triumphs over Sauron, and re-establishes the Kingdom of Men which stretches both north and south.
So Lindariel, I don't think Gil-Galad's fostering of Elros and Elrond was for the purpose of grooming a High Noldoran heir for Middle-earth. The glory of Gondolin was gone, and I don't believe Tolkien ever entertained the thought of carrying on the Noldoran lineage. The Elves' days in Middle-earth were numbered. Elrond's purpose in the story was to be the minstrel and counsellor to Gil-Galad and subsequently be the caretaker of the Elven sanctuary and refuge...The Last Homely Home...through the final battle with Sauron, and until the Elves left Middle-earth to Men.