Does a seatpost affect the rigidity of the frame? Is it possible to use a zero offset seatpost to make the bike frame more rigid?

I consider the Crank Brothers Cobalt 1 Straight Seatpost.
Seatpost setback is used to get the saddle in the right place to get proper rider fit on the bike. The amount of setback required is dictated by the desired riding position, the seat tube angle and the bottom bracket to saddle distance.

If you switch out a seatpost that has some setback for one without you are either going to mess up your riding position or be forced to mount your saddle to far rearward in the rail clamps.