Would a pure flying wing in straight and level flight have zero wing bending stress?
Net wing loading (weight per unit lifting area) can't be zero in level flight.  Lift must equal weight, else the aircraft will accelerate in the direction of the net force.

The loading distribution on the Helios is designed to reduce flexing moment along the span, which is an entirely different thing.  As you note, that optimization wasn't perfect, although pitch oscillation wasn't necessarily related to the wing flex that shows in the video.