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Again I had a run-in with the Kuzari "proof" (or "principle") that Judaism must be true. In a nutshell it goes like this (you can read a longer version at Aish: Did G-d Speak at Sinai or Simpletoremember: A Rational Approach to the Torah’s Divine Origin):

G-d appeared to the entire people on Mount Sinai. The story has been passed down through the generations. Nobody could make up such a story because the people involved would not go along with a lie, they had been there. So the revelation must have taken place and the Torah is true.

With this post I don’t want to say that Judaism is not true or that G-d does not exists. I truly believe in G-d and I believe that Judaism is the best way to connect to Him (at least for me). But I also think that it is a question of faith and that nobody can prove G-d or the truth or falsity of some religion. And that it is dangerous to build faith on top of one such "proof". Here are a few claims this "proof" makes and short attempts at refuting them. If you have more, write a comment.

Claim: This is the only story of such mass revelation in all religions.
There appear to be mass revelation stories in Aztec mythology, several pagan religions and hinduism.
Also some of Jesus’s miracles were watched not by millions but at least by large groups of people (e.g., giving bread and fish to 5000 people at the lake). So it is certainly possible to claim that some event with a large amount of witnesses happened even though it didn’t (sorry, Jewish blog).
Mohammad once split the moon (allegedly, sorry, Jewish blog), that could have been observed by all people on the planet.

Claim: The people involved wouldn’t believe it if you made such a story up about them.
So this is the situation where one person says "hey, remember, yesterday we all heard G-d speak to us".
Maybe most people really didn’t believe. Maybe of the people spoken to most left and only a few believed and stayed. These few then gave the story to their children as truth. They would still claim "it happened to all 3 million" because they actually believe it did happen and those who left only denied it happened. And they might conveniently forget that most didn’t believe and left the group.
Maybe even those who stayed didn’t all believe the story but played along because of social factors (family, friends, power position) or coercion (stoning people who work on Shabat etc).
In short, there are several ways of making group of people say that something happened to a larger group of people including them.

Claim: You cannot make people believe they were present for something that did not happen.
"The Miracle of the Sun was an event which occurred just after midday on Sunday 13 October 1917, attended by some 30,000 to 100,000 people who were gathered near Fátima, Portugal. Several newspaper reporters were in attendance and they took testimony from many people who claimed to have witnessed extraordinary solar activity." (Wikipedia:Miracle of the sun)
Unless you claim that this miracle really happened (btw it was officially accepted by the Catholic church in 1930) – yes it is possible to get many eyewitnesses for something that never happened. Probably some unusual sun activity took place and the rest is wishful thinking.
How this may work for Sinai: The text talks about fire, smoke and thunder and that it was very terrible for the people who witnessed it. It is conceivable that a group of people comes upon a vulcano, comet or some other natural catastrophe-type event, they all have a horrible time but are amazed at the powers, faint and then afterwards one charismatic leader says that was G-d speaking and he said XYZ.
So here nothing really happened but you have a group of people who really believe something happened.

Claim: There were 3 million people involved in the revelation.
We only have the Torah’s claim that there were this many people. It could have been the story of one family or clan who claimed there were more people around (see above).
Or even they claimed it happened only to them and the story was then adopted by other clans. If we look at it as some conversion-like acceptance of the original clan’s way of life it is absolutely realistic to assume they passed the story on as happening to their own ancestors and so it became more and more people that supposedly should have been present.
Or maybe the story was passed on as "happened to our ancestors" and at some later point in time the numbers were added (and chosen to make it more impressive).
In any case, how many people really left Egypt and were at Sinai is a different question for a different day. But the exact number does not really matter that much here.

Claim: Such a claim would be easily verifiable, nobody would make it up.
We are talking about an age without modern means of transportation and communication. How verifiable is it when a group of 10 or 20 comes and says "we are part of this larger group of 3 million (who are currently unfortunately a year’s worth of travel away or they all unfortunately died in the desert or …) and G-d talked to all of us"?

Claim: You cannot say at a later point ‘your ancestors were there’, you would not believe this if your family does not have such a tradition.
Apparently this works, check 2nd Kings 22/23 where king Josiah finds a Torah scroll and they seem to have forgotten everything in it (including minor details like Pesach) and then they go on to re-institute Torah law. Nobody seems to come and say "hey I didn’t hear that from my parents".
Another example are midrashim, handed down as accurate oral tradition. Many of them talk about experiences witnessed by the whole people such as during the exodus. Many of them contradict each other, so they cannot all be true. Nobody said "this is not what my father told me", they have been recorded and handed down to us.
It might be hard to introduce a completely new story of this magnitude, but here the story might build on an existing story like "we all heard from this prophet who spoke to G-d on the mountain" and modify it over time slightly from adding "and we all stood at the mountain, too" to "actually G-d spoke to all of us directly" over a long period of time.
Also, remember that people were for the most part illiterate, more superstitious and less analytical as we are today. And even today people can be made to believe all sorts of things that alledgedly happened to their forefathers. Ask Egyptians about the Yom Kippur war for example, you’ll get a truly amazing story of a war victory. And the generation who actually lived during the war is still alive.
King Arthur is also brought as an example for a figure who probably never existed but is taken as a real identity-building figure of history by many.
Maybe the story was introduced originally as a myth and everybody knew it was not true, but they retold it as if it were true anyway (after all, it’s a great story). People in ancient times did not have the same idea of history = objective truth than we do today. It was common to exagerate or tell stories to make a point, objectivity was not the main value. So the story was passed on originally as myth and only at some later point the literal understanding of the story took over.

Claim: This experience was a key element of faith that was passed on faithfully through all generations until today.
A few days after this experience the Israelites went back to worship a golden calf. All over Tanach are examples where the Israelites go astray and G-d needs to send prophets to bring them back on the right track. And the revelation basically consists in G-d saying "I am the only G-d, don’t worship others". How memorable can the experience have been? How faithfully has it been transmitted to the future generations?

Claim: All our ancestors heard the whole Torah at Sinai.
Actually the text says they only heard the first part and then couldn’t handle it any more (Exodus 20). It is actually pretty unclear what exactly they heard (which by the way is in itself curious). Most people say it’s the first verses, some say it is the whole 10 commandments. Nobody I ever met says they heard the whole Chumash (books of Moses). Interpretations that they heard only a "loud voice", but no discernible words, are also consistent with the text.
Even if we could prove with absolute certainty that this one event of mass revelation happened and they heard verses XYZ – so what? This says nothing about the truth of the rest of the Torah, all the do-s and don’t-s and the interpretations over centuries. Even if you say that Moses received the whole written and oral Torah at this point, it was still only Moses, one person, receiving it. The people heard maybe a voice, maybe 2 verses, mabey the whole 10 commandments. This doesn’t automatically lead to all of Judaism being true.

Again, I am speaking out against using the above claims as a "proof" of the truth of Judaism. People will either believe or not. But if their belief is based on a falsehood like this "proof" it will shatter the moment they are faced with a counterargument. Better tell them that faith cannot have proof. That this "proof" does not really proof anything has nothing to do with me believing that the Torah is true or that G-d exists. Let people appreciate the beauty of Judaism and of G-d, build a connection and then they can decide for themselves to believe.

A few links if you want to further read about this topic:
Mark Perakh: Dreaming Up… – Israel in Sinai
Baruch J. Schwartz: What Really Happened at Mount Sinai?
DovBear: Demolishing dumb arguments (The mass revelation argument for the Torah’s Historical Veracity)
Larry Tanner: Definitively Refuting the Kuzari Principle (also see the Index of Kuzari posts and Kuzari Principle Round 3 for more)
Skeptic but Jewish: Main Argument for Judaism: Refuted
Torahphilosophy: Truth of Judaism (read the comments)
Sam Lebens: The Kuzari Principle
Orthoprax: Aztecs National Revelation II
FailedMessiah: The Kuzari Proof, The Exodus And Passover