edited for clarity
The Kuzari argument is essentially that a lie of national revelation at Mount Sinai is too ambitious of a lie to ever gain traction. All other revealed religions trace revelation to either a single individual or a very small set of individuals. The national revaluation was asserted to be witnessed by 2.5 million people and established a functioning legal system that was practiced as far back as we can historically verify. Further, the written torah was disseminated as a matter of law to individuals and practices specifically enumerated as “remembrances” were practiced as well. Put another way, “what was the topic of conversation at the first passover seder?” While there are certainly hypothetical situations in which we can have doubts, as with literally all historical inherently unfalsifiable occurrences, in light of the Kuzari argument, is it reasonable to believe it happened?
Again, I want to plant the goal posts firmly. I'm not saying irrefutable. I'm saying reasonable to believe vs. the alternative. Is it possible Aliens built the pyramids and falsified Egyptian history? I mean, it's possible. It is less reasonable to believe than believing the Egyptians did it. Can we prove aliens didn't build them? Well, no, but come on. What is the rational basis for believing it and disbelieving the alternative? Is it more reasonable to believe aliens built them or that egyptians did? With that in mind, my defense of the Kuzari argument.
There is a legal concept called a self authenticating document. This is a document in which the facts were not in dispute at the time of the drafting of the document, and at no point in the chain of custody is there suspicion of alteration since then. Legal documents and newspapers are prime examples of this because they are authoritative at the time they are drafted, there are no objections at that time, and they can be verified to be free of corruption.
Imagine that you wanted to establish reasonable belief in the existence a child named Billy who appears on the school rosters, he's always been on the rosters, but nobody at the school can remember a Billy ever being there. However, they have a document. Simply by having the document, can we say Billy existed? Maybe, maybe not. Certainly not all documents could do so. Let's say the document is a court order. Let's say it says, “Having had Billy before me today, and on the sworn affidavits of nursing staff, it is so found that Billy is the natural child of Claire, a US citizen, and is therefore entitled to an American Education. It is therefore ORDERED that Billy's name will not be removed from the roster.” Further, a gold seal copy of the order with the Judge's name and signature was found in the court archives. Is it reasonable to believe Billy existed at least at the time of the Order? I think we can say uncontroversially yes.
So it is reasonable to accept the testimony of documents that have been kept free from corruption and they were accepted as true at the time they were drafted. The torah is alleged to be the functional constitution of the Israelite government, and is alleged to have several chains of custody. There are no serious attacks on the chain of custody. All available manuscripts of the torah, even though they have minor differences, agree on the basics of the story. Also, DH does not have any archeological support nor does it get around the question of the origin of the story of national revaluation even if true. The Kuzari argument at this point boils down to, at no point between the revelation at Sinai and now does it sound reasonable that someone would be able to pull off the lie, “the Torah was given to your direct descendants, they have been meticulously preserving it for centuries, it is the basis of our existing legal system, G-d revealed himself to every man, woman, and child, and this is the first time you're hearing about it” make sense.
Chumashim (bibles not used for rituals) written in the script of the first temple are preserved, and the Samaritans broke off during that time too, were hostile, and unlikely to adopt the religion of an enemy people. The last time that Israel and Judah were a unified entity and culture was the Unified Kingdom under Solomon. That allows for less than 500 years, between the revelation at sinai and the breaking up of the kingdom that one can start the lie and spread it across all Israel. Given the ridiculousness of the lie, it does not seem plausible for the lie to gain such traction.
While it makes sense that a few people would believe the lie, it is too easy to correct it against the collective memory. Oral histories have an error correcting mechanism. Each time someone gets a transmission of the oral history, they can immediately check it against the majority of people they are surrounded with. Nobody is giving one person a story, sending them off for a hundred years, and then checking it against the story of another community. What is being alleged here is a tight knit community that is constantly transmitting the story to new people raised in the community, and details are constantly being checked against the majority.
Imagine if someone came to you and told you the George Washington Story, except instead of chopping down a cherry tree, he chops down a pear tree. Instantly, you would correct him, and then you'd go ask a bunch of other people to back you up. When it's 99:1, he will relent that he must have been mistaken. So when the next group tells the story, everybody agrees it's a cherry tree. The first time someone hears the story concerning Sinai, they're going to go tell a friend. They're going to say, I asked my parents and they never heard of it. Instantly, the story is refuted and cannot gain traction.
Now, we know the George Washington story is false. That's why this is a two pronged argument. We can trace the transmission of this story back to a single individual, Mason Locke Weems. His biography is not the type of document that could be considered an authenticated document. It's clearly hagiography and was never accepted as true. Further, it's a very easy lie to pull off not at all analogous to the revelation at Sinai. However, what is extremely important is that once this story enters the collective consciousness, it cannot ever change. If Mason Lock Weems said he chopped down an oak tree, it would forever be an oak tree. To change the hypo, let's say the question is “did Napoleon exist?” We all have a collective “memory” that Napoleon was French. We can trace this back to authenticated documents that document the occurrences in the wake of the French Revolution. Even if we didn't have the original manuscripts, we would have copies of those books and the collective memory that this is exactly what happened.
For another example, we know that Socrates was generally considered to be a Sophist by the Athenians who were not his immediate disciples. How do we know this? Well, we have Plato testifying that was the case, Xenophon testifying that to be the case, and Aristophanes really hammering that point home. So we have multiple chains of transmission. Further, we have a document, Aristophanes “The Clouds” which is explicit on this fact, and though we don't have the original document, we are certain that it was an accurate perception and that what we have is basically right on the basis of multiple chains of transmission and general agreement.
The revelation at Sinai lie is simply too ambitious. Compare to the revelation claims of every other religion. Christianity? Only a couple people can verify the resurrection. You simply have to accept it on their authority. Mohammed? A man alone in a cave. Nobody can verify that either. You either believe them or you don't. In all these cases, once the story gets going, it is unchangeable. But right at the moment of the initial lie, there's no independent method to verify. National revelation though is verifiable. You can instantly go check to see if its true at the time the lie is proffered. “Did anybody else hear about this huge event that started our government and gave us our freedom? No? Okay then.”
So given we have a self authenticating document, it was not in dispute at any time since then, and we have a continually functioning legal system that claims it as its constitution, it becomes extremely reasonable to believe it. Certain knowledge is going to be impossible. Historic occurrences are not possible to verify. However, given the above, it is reasonable to believe it occurred because it is more likely than not a fabrication would have been immediately discovered and corrected.