This question is more-or-less answered in The Nature of Middle Earth in a way that seems satisfactory - to me anyway: the first existing Elves were not leadership material and mostly ended up Avari, the unwilling who never left Middle-Earth.
Fairly clearly then Ingwe, etc. are not "First Elves." What then became of older generations?
This can be got over. The Quendi at first (to 3 gens.?) were very philoprogenitive. They mated almost at once with their predestined mates. It was not for some time, when their young, inexperienced fear began to take command, that their other faculties demanded fulfillment, and they began to be absorbed in the study of Arda. The younger generations therefore progressed rapidly in strength, nobility, and intellectuality of character, and made natural leaders. The first few generations (expending much vigour in begetting) were least adventurous and were nearly all Avari in the event.
In another draft in the same book, trying to lay out an early timeline, Tolkien specifically mentions Tata, Inel, and Imin as Avari:
The "Ambassadors" [Ingwe, Finwe, and Elwe, all of the 24th or 25th generation; only the youngest were willing to go and these three were elected from among those] return. Great Debate of the Quendi. Imin, Tata, and Inel are ill-pleased, and regard the affair as a revolt on the part of the younger Quendi, to escape their authority. None of the First Elves (144) accept the invitation. Hence the Avari called and still call themselves "the Seniors."
Tolkien does also consider sending the Three to Valinor with Ingwe, Finwe, and Elwe, but the younger Elves end up being the most persuasive and passionate speakers, and this draft has Enel not at all impressed with Valinor. But it seems that Tolkien, when he reflected about this at all, tended to go with the "Tata, Inel, and Imin were decidedly Avari" route: that narrative crops up more than once in The Nature of Middle Earth.
It also happens that Elves did typically resign kingship to their descendants after a period of time, we just don't see it happen much in-story because from an Elvish perspective, the recorded history we have of Middle-Earth is relatively short.
Secondly, in any case: Elvish lords or Kings (as Numenoreans later) tended to hand on lordship and affairs to their descendants if they could or were engrossed in some pursuit. Often (though we don't see it in Beleriand, since the War occupied so short a span of Elvish-time, and the lords and Kings were so often slain), after passing 200 age-years [NB: these are not the same as our years] they would resign. It would thus be young, eager Quendi of some later generations (whose fathers or grandfathers were lords) who were chosen and/or willing to go as ambassadors to Aman - after which they would be preeminent and obvious leaders of the March. "The light of Aman was in their faces", and other Elves were in awe of them.