Earlier quoted context omitted.
i think you're confusing collecting area with airflow cross-section. they're completely different things, unless somehow the air is blowing through the side of the (solid) bottle.
Not really. This volume of air has to make contact with the bottle, whatever the collecting area's shape. Imagine loosely fitting a tube around the neck of the collecting area. All the air has to pass through that tube.
you are calculating the swept volume. but only a small fraction of the swept volume "makes contact" (ie is roughly within the mean free path of a water molecule) with the collecting surface (which appears to be necessary here for water to be extracted).
with netting your argument is closer to being correct. but this is an impervious bottle, not a net.
if you implemented the tube you suggest (and i understand that was simply an illustration to explain swept volume) then pressure would increase in the tube slowing flow and/or flow would be predominantly around the edges, with, again, a stagnant higher pressure region in the centre.