"One of the major problems with the Kuzari proof is that there are two bottlenecks in traditional Jewish history where the Torah was re-introduced, one during the reign of Yoshiyahu Hamelech (King Josiah) and one during the time of Ezra.
Both times, the Torah was "re-revealed". So the continuous narrative that the Kuzari proof depends on does not exist, according to Tanach itself.
Even if you assert that what was found in Yoshiyahu's time was just Devarim, and the people knew the rest of the Torah, that claim is harder to make in the time of Ezra, These were returnees to the land who had clearly been living without a Torah for some time, or the returnees who came back with Ezra. So the revelation story was new to them, yet they believed it, just assuming that the chain had been broken. According to your reasoning, they should have said "wait! my parents and grandparents didn't tell me that!" Instead, they just assumed that in the upheaval of exile, the tradition had been lost."
The return to Israel of some 42,000 Jews from the Babylonian Exile is regarded by Bible critics as a watershed moment in Jewish history when the Torah (since presumed forgotten in the 70 year exile) was reintroduced to the people. Critics like James Kugel make quite a lot of hay with this assumption - in his case constructing an entire alternate universe of Judaism.
Taking a peek around the book of Ezra I found a few items that would seem to be a bee in the bonnet of our reader's hypothesis that the "tradition had been lost."
- In Ezra 3:4 it says the the returnees from Babylon kept the "Feast of Tabernacles" (Sukkot). Sukkot is a complex holiday with a lot of different parts and a lot of mitzvot (commandments) that cannot be kept without extensive knowledge of Jewish law and tradition. If it was lost they would be unable to keep it.
- In Ezra 3:6 the people make offerings on the "first day of the 7th month" (Rosh HaShana). Thus they knew both about the holiday and the laws and traditions associated with making offerings.
- In 3:7 they begin to build the Second Temple in Jerusalem indicating that they knew about the historical significance of the city as well as the many complex laws of the construction of the Temple and its vessels.
- In 3:13 some of the older people (who remembered the First Temple) began to weep - noting that the Second Temple paled in comparison to the First and indicating to us that there were people there who were alive from before the point that the tradition was supposedly broken.
In short, the tradition, while it may have been compromised during the exile, was not lost and the notion that the Sinaitic Revelation was forgotten among the populace or somehow reintroduced by Ezra or others just doesn't hold water.