Intro:

You go to a ‘learning about the faith weekend’ or some such event expecting to be enlightened.  Up to the podium steps a bearded Rabbi whose face speaks of wisdom and learning.  Eagerly you wait for the inspiration you’ve been waiting for to believe.  Out of those lips flow an argument which is akin to a method of argument you’ve learned about in calculus class: induction.  Starting from a base generation which witnessed an event at Mount Sinai, this story has been passed down through the generations unchanged until it reached your ears.  A miracle!

The Argument:

Numbers 1:24-26 tells of 600,000 adult males that witnessed the giving of the laws to all at Mt. Sinai.  Since it is impossible to fake the simultaneous testimony of 600,000 adult males (approx 2-3 million all together) the story must be true.  Since your (Jewish) ancestors witnessed and passed on the story of God publicly giving us his laws, we must obey his laws and worship him.

Another plausible solution:

20,000 people witnessed a volcano and worshipped it.

20,000?

There is a confusion in hebrew between alluph (chief, master, family clan) and eleph (thousand).  They are spelled the same way since the torah (and most semitic languages) omit vowels.  As a result a redaction or copy error has led to the interpretation of alluph as eleph.  600,000 adult males in ancient egypt is just a historically ridiculous number.  For more info, visit:
reconciling-biblical-numbers-three-million-at-sinai-is-making-a-mountain-out-of-a-molehill

Volcano?

Many descriptions of Mount Sinai sound more to me like a volcano than anything else.  The fact that the Israelites heard God’s voice in this may simply mean that they were frightened by the volcano and thought it was God’s voice?  Recall this is the time of sun worship.  It was common then to worship natural phenomenon and deify them.
See:
http://en.wikipedia.org/wiki/Biblical_Mount_Sinai#Biblical_description

Do I really believe Israelites worshipped a volcano?

I believe that the Israelites may have heard loud scary sounds at a volcano or some other natural phenomenon.
See, second part of:
http://extremegh.blogspot.com/2008/07/apology-to-rjm-chazal-destroy-kuzari.html

My point is that my alternative explanation is as plausible, if not more plausible than the Kuzari explanation.  Hence the Kuzari proves nothing.

Why should we care about debunking the Kuzari Proof?

“Every time we let ourselves believe for unworthy reasons, we weaken our powers of self-control, of doubting, of judicially and fairly weighing evidence. We all suffer severely enough from the maintenance and support of false beliefs and the fatally wrong actions which they lead to…. But a greater and wider evil arises when the credulous character is maintained and supported, when a habit of believing for unworthy reasons is fostered and made permanent.” — W. K. Clifford

That’s why!