This is a great question, Angelimir! Here’s a long-winded answer.
If I recall correctly,
Galadriel is described at one point as the greatest of the
Noldor save
Fëanor alone. (This was in the context of his asking her for a single strand of her hair, which she denied him, so that they were
unfriends forever; hence the shock of the
Elves when
Gimli named a single strand of Galadriel’s hair as his desire at
the Nine Walker’s departure from
Lórien.) Galadriel is extraordinarily powerful, even for a Noldo; however, she is not a war-leader, and the whole purpose of the Noldor’s coming to
Middle-earth was to wage war upon
Morgoth, at least for everyone but Galadriel. Her purpose was, perhaps*, more because she wanted a kingdom or
realm of her own. She went to Eregion with
Celebrimbor, the chief and perhaps only surviving descendent of Fëanor in
Arda, but Celebrimbor and the Mírdain (Elven-smiths) cast her out after
Annatar,
Sauron in disguise, cozened them with his gifts of knowledge that he truly possessed, but misused to their ruin. Galadriel together with
Celeborn eventually ruled
Lothlórien, but that eventually ended with the destruction of the
Ruling Ring.
All that said, I think the answer to the question,
Why did Galadriel not rule as High Queen of the Noldor? is that she was not a warrior. The
Númenóreans changed their law of succession from agnatic primogeniture (succession by the eldest male descendent) to simple primogeniture (succession by the descendent) only after six long generations of peace lasting more than 900 years. Their reasoning was that they were unlikely to be attacked in a way that drew their sovereign into battle. (I think there is reasonable evidence that Tar-Telperiën, the second Ruling Queen, made her nephew Minastir co-regent with her in order that he could prepare the Númenóreans for their war in alliance with the Eldar against Sauron: he served as the warrior-king and rightly received credit for sending the expedition to Middle-earth, while Telperiën continue to rule as the senior sovereign. In
Gondor, this arrangement between Narmacil I and his nephew Minalcar, who later ruled as Romendacil II, saw Minalcar bear the title
Karma-kundo, or “Guardian of the
Helm”.)
As for
Elrond, the situation is a little different. He was not technically a Noldo, but a Sinda. It’s easier to understand if we first examine this from the perspective of his brother,
Elros. Both of the brothers were Half-elves. Choosing to be mortal, Elros was counted as a Man, and was accounted the Chieftain of the Third House of the
Edain, which he inherited from his father
Eärendil, who inherited it from his father
Tuor, who inherited it from his uncle
Húrin after his cousin
Túrin Turambar, Húrin’s son, died without issue. Note that Túrin never claimed to hold that position, Húrin held it while long a prisoner of Morgoth, but Tuor, apparently, was recognized as the leader of the
Men of the Mouths of Sirion, along with Elwing, the granddaughter of
Lúthien and
Beren (last chieftain of the First House), whom Eärendil later married, uniting those two Houses of Men into one line; and since the Second House, which was nearly annihilated, chose its Haldad or chieftain from among the kindred of the leading family, their survivors were also free to follow Eärendil and later Elros, since they were (apparently) the only survivors of that clan of the tribe. Finally, the
Kings of
Númenor never actually referred to themselves as the “
House of Elros”, but rather as the “House of
Eärendil”, harkening back to the position of their forefather as, first and foremost, leader of the Third House of the Edain, or at least what was left of it. Note that the position need not have been
claimed: Elros was
de facto chieftain of the Third and First Houses
by right, by birth, and no doubt by accomplishments.
What then of Elrond? As Elros was counted among Men, so Elrond was counted among Elves. His mother Elwing was a Sinda of
Doriath, daughter of the second and last recognized King of Doriath and High King of the
Sindar, Dior
Eluchíl or “Elu’s Heir”. His father Eärendil was descended of
Turgon, reckoned High King of the Noldor, but through his grandmother, Idril.
Technically his patrilineal line marked him as a Man, but since as a Half-elf he’d chosen to be counted among the Elves, Elrond was one of the Sindar because that’s what his mother was†, and he had chosen his Elven heritage.
Elrond never accounted himself equal to
Gil-galad, the last High King of the Noldor in Middle-earth, though both of them were born in Middle-earth; instead he served as Gil-galad’s herald and trouble-shooter, sent by the king to Eregion to bolster its defense against Sauron in the War of the Elves and Sauron, leading the survivors of that realm to safe refuge in the
Misty Mountains when Eregion was overrun and destroyed. There Elrond remained, ruler of his own little sovereignty for over 4700 years.
As Olorgando pointed out, there were fewer and fewer Noldor left in the Second and Third
Ages. Most of them returned to
Aman at the end of the
War of Wrath at the end of the
First Age, and more emigrated in the following centuries. Many were killed in the
wars of the
Second Age against Sauron, the War of the Elves and Sauron and the
War of the Last Alliance. Over what polity would Elrond claim sovereignty? Eregion had been a mixed population of Noldor and Sindar, but it was gone. Likewise
Lindon was mixed; and so were the survivors of the wars of
Beleriand at the end of the First Age near the Mouths of Sirion.
By the way, Elrond’s declining to accept or claim a royal title had a powerful influence on the surviving Dúnedain of Arthedain: After the death of Arvedui following the conquest of Fornost by
Angmar, the leaders of the Dúnedain in
Eriador called themselves “Chieftains” rather than “kings”. I think this choice was influenced by Elrond’s lack of any title greater than “Master of
Rivendell.”
───◊───
* “Perhaps” Galadriel returned to Middle-earth to found her own realm to rule. The textual history of the history of Galadriel, as Elthir can tell you better than I, is most complicated.
† Elwing was also Half-elven, though. Dior Eluchíl was mortal because both Beren and Lúthien were mortal when he was born. The Elves of Doriath probably never knew this: Dior died in the Second Kinslaying of the Noldor, when the Sons of Fëanor sacked Doriath in their attempt to obtain the Silmaril of Beren and Lúthien. Elwing’s mother was an Elf.