Many aspects of our mythos are not generally known: the uncountability of the reals, the classification of surfaces, the approximation of waves via simple harmonic oscillators, the infinitude of primes, the conformal mapping theorem, etc. etc. In fact, most of what we take to be well-known is unknown to the layman. In the 1960s Time-Life books, and Bell-labs and other companies popularized various topics.
Now independent producers are doing the same. Vi Hart comes to mind, mathematical artists, and yes youtubers and math rappers :D So the desideratum of Daniel may be coming about through the new media.
It might be better to be on Discover or History if only if late at night. It would certainly be better than the ghost and paranormal stuff --- Wait, how about para-compact stuff!
Instead of thinking how good it would be if we did this, then do it. Two people, a 200 video recorder, and a tripod is all that you need. As you get better, add production value.