Sharp over the entire cyclist, obviously background is falling out of focus due to that thing we call depth of field

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Does it provide corner to corner sharpness in a flat field?
Yes barrel distortion is pretty bad. As far as pincushioning goes, that's a telephoto effect.
Barrel and Pincushion distortions are the result of the lens computation decisions taken by the designer. They can effect wide angle and telephoto lenses, zooms usually pass from one to the other being neutral in the centre of the range.
Distortions are unbelievably simple to correct, however perspective distortion is not.
Not so... The lens in question seem to suffer from moustache distortion. this is very difficult to correct indeed.
Perspective distortion is the the simplest to correct of all as it is geometric.