I may be feeding a troll but as I'm finding it difficult to get to sleep...
Not entirely true as there was fibre insulation between the inner and outer walls of the spacecraft which acted as a fairly effective shield to most particle radiation but the decision not to apply any further shielding had nothing to do with the Van Allen belts and was a calculated risk by NASA who were more concerned with the lethal danger from either a powerful burst of cosmic rays (relatively rare) or an extremely violent solar flare event - more common but usually linked to sunspot activity which has a roughly cyclic nature and therefore low activity periods can be cautiously predicted to minimise the risk.
As
@aguero93:20 mentioned where the Van Allen belts had to be traversed it was done at high speed to minimise exposure plus the flux levels of the most dangerous radiation are relatively low and not uniform throughout the belts.
Also if you believe the earth is a disc, how are the Van Allen belts distributed around the disc? In the case of a sphere with 2 opposite magnetic poles the belts form as doughnut-shaped energy fields. In fact I could argue that your acceptance of the Van Allen belts' existence and the science involved in establishing that rather contradicts the vid's core argument.