I like the thought that they were never lost. We can call them lost because we do not know where they are. But I do not think they were lost. A small tribe traveled north. They did not go with the majority to the south. They split in two to make the odds of surviving better.
Actually, the majority of the Israelites were lost in the Assyrian exile. The Southern Kingdom was made up of only three tribes: Judah, Benjamin, and the majority of the tribe of Levi. The Northern Kingdom was made up of the other nine of the twelve tribes, plus a minority of the tribe of Levi: and of all the twelve tribes, Ephraim-- the dominant tribe in the Northern Kingdom-- was the most populous.
The two kingdoms did not voluntarily split to increase their odds of survival: the split between the two kingdoms predates the conflict with the Assyrians, and stems back to political infighting about centralization of authority and religion, quarrels over taxes, and the conflict between henotheism and monotheism. The Southern Kingdom believed that both government and religion should be centralized in Jerusalem, that the king had the right to impose taxes in order to support large national projects (such as building the Temple and royal palaces and so forth), and they leaned ever more toward monotheism. The Southern Kingdom was ruled without interruption by the Davidic dynasty. The Northern kingdom believed that government should be tribal, with the king holding authority relatively loosely, and with little authority to levy taxes; and that religion should be practiced at many shrines and slaughter-sites scattered over the land; they were not ruled by a dynasty, but by a succession of royal houses, whose ascent to the throne was often due merely to military prowess; and they were deeply henotheistic, worshipping many other gods alongside YHVH.
There is very little evidence, either textually or archaeologically, that the lost tribes did not, in fact, vanish into assimilation.
The ten "lost tribes" of the northern kingdom were not lost in the exile, for they are "found" in the rest of the Bible.
1) in prophecies relating to the return from exile:
---Jer 3:18, 31:27, 31 (where "Israel" is the northern kingdom of the ten tribes)
---Eze 37:15-22 (where "Ephraim" is the norther kingdom of ten tribes--Isa 7:17)
2) return from exile: Ezra 6:17
Of these, the prophecies of Jeremiah almost certainly do not mean the lost tribes. When Jeremiah uses the phrases
bet Yehudah im bet Yisrael "the house of Judah and the house of Israel," he means it metaphorically, although some say he means the tribe of Benjamin. Yet even if he did mean it literally, Jeremiah lived less than 100 years after the Assyrian Exile, and he may have had reason to hope that the exiles might yet return under the proper conditions (which he believed to be the complete embrace of monotheism, and the proper worship of YHVH by all the Israelites). By the same token, Isaiah lived during the era of the Fall of the Northern Kingdom, and would have had great hopes that the exiles might yet survive and return home. These events were not historical or messianic for Isaiah and Jeremiah, but (loosely speaking) current events.
Ezekiel is a messianic prophet, who lived during the Babylonian Exile of 586 BCE, over 200 years after the Fall of the Northern Kingdom. His prophecies are, among many other things, an origination point of a school of messianic thought that believes that the messiah, when he comes, will be able to sort everyone's proper bloodlines, and any survivors of the lost tribes who have somehow remained Jewish will be able to reclaim their lost identity and return to the fold.
But there seems to be small chance of such a thing occurring. It is unlikely that any further lost Jewish communities will be discovered, and if they are, the vast majority of such communities found went into exile in much later times, long after the Assyrian Exiles-- it is deeply unlikely that a community could have preserved its Jewishness for over three thousand years and yet remain undiscovered in the modern world.
It is much likelier that Ezekiel's prophecies also were either conditional or were metaphorical or allegorical, using imagery not intended to be taken literally.
However, with Ezra, when he mentions all twelve tribes, this is because it was the custom in the Second Temple to still offer expiation for all twelve tribes, and to offer sacrifices in the names of all twelve tribes, even though the other tribes were long lost, and it was deemed that they would be lost until the end of time. They kept offering sacrifices and offering expiation for all twelve tribes as a memory of those lost, and because the Covenant was made with all twelve tribes, and most importantly, because a very small number of the Israelites of the Northern Kingdom had escaped the Assyrian Exile by fleeing to the Southern Kingdom, where, by the time of Ezra and Nehemiah, they had mostly forgotten or lost their tribal affiliations, as the Jews of the Second Temple period mostly lost tribal identity, except for the tribe of Levi, who were the priests. Therefore, sacrifices and expiations were offered in the name of all twelve tribes, because it was theoretically possible that the community contained some few individuals with descent from each of the twelve tribes, even if those individuals were unaware of it.