Canon EF 70-200mm f/2.8L IS II vs Cheap Tamron on 70d (autofocus, image quality & stability)

Comparing video performance of the Tamron AF 70-300mm f/4.0-5.6 SP Di VC USD verses Canon EF 70-200mm f/2.8L IS II USM on Canon EOS 70d at 200mm, which is equivalent to 320mm on a full frame camera. Obviously, the Canon EF 70-200mm f/2.8L IS II USM has a fixed aperture of f/2.8 throughout the zoom range and will perform much better under low light conditions.
Video was shot on a monopod.

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I thought that the Tamron 70-200mm lens had troubles auto focusing on the Canon 70D?