Nahmanides in his commentary on Deuteronomy 4:9, regarding the transmition of the divine revelation at Sinai:
And the purpose of this commandment is very great, because if the words of the Torah came to us from the mouth of Moses alone, even though his prophecy rang true through signs and wonders, if a prophet arose amongst us or a dreamer of dreams and would command us opposite of the Law and gave us a sign or wonder, doubt would enter the hearts of the people. But when the Torah reached us from the mouth of The Might to our very ears, and it is our eyes that see there is no intermediary, we will disregard any dissenter and any doubter, and find him to be false. And no sign will help him and no wonder will save him from death by our hands, for we know his lies.
This is what it means when it says 'And also in you (Moses) they will forever believe', because when we tell our children what we saw, they will know that the matter was true without a doubt, as if all generations had seen it, for we would not testify falsely to our children and we would not bequeath nonsense that has no purpose. And they will have no doubts in our testimony, rather they will certainly believe that we all saw it with our eyes, and all that we told them...
So while this is certainly not a logical proof, it is one that relies on something that has in the majority of cases always existed in humanity: a parent's love for their child. So we trust our parents about what happened at Sinai, who trusted their parents, who trusted theirs, etc... who trusted their parents who were physically present at Sinai and who personally experienced the absolute existence of Gd, along with the rest of an entire nation, which could have been upwards of 3 million people, with no middle man.
Yes there have been breaks in the chain in many individual family trees, but for every family generation that was lost, thousands more were maintained. Thus, as a nation unit, we span the gap. Like a screen door, that can have holes in many places and still stretch the length of the frame.
This parent-child lattice, spaning generations of families that began with a nationwide revelation would thus convey the authority, the duty, and the covenant of the Jewish Religion, and all that it implies.