Following the path set by R' Yehuda Halevi, Rambam and more:
Our tradition is completely oral. Very few things were (initially) let to be written (תנ"ך), the rest kept alive by oral tradition.
As R' Yehuda Halevi writes in Kuzari - the Torah doesn't prove itself. The basis of Torah isn't the Torah. The very fact that is was passed down through the ages by many reliable people is strong evidence - as good as hard proof. That's why we believe it's true, and then we rely on all written. Otherwise you have sort of a paradox, as your question points out.
Besides - how would proof help? Even participating in a miracle doesn't mean you get everything straight from there on (see Ma'amad Har Sinai & golden calf) what would a video clip do better?
In Shmuel I chapter 7 it says: וינהו כל בית ישראל אחרי ה, the whole nation was yearning for G-d.
Real belief comes from hard work of education - והלך מדי שנה בשנה... ושפט את ישראל - he would go round every year and judge (generally "judging" also meant teaching Torah) the people (ibid).
Many people close their eyes to simple facts of reality. Even today there are people who have been through miracles (though not as explicit) and do not see the hand guiding these happenings.
Remember Pharaoh after 10 miraculous plagues leaving no doubt of G-d - chasing Israel into the sea. May none reach that level of blindness.