Oversized tubing is the key to stiffness. The old Vitus frames = flexy. Cannondale's are known to be stiff.....they have oversized tubing. Tubing sitffness increases in multiples as the the diameter increases. Diameter has more affect than wall thickness.
But think about what you are implying there - hypothetically if we could build a frame of some new design with some unobtanium new material such that it had absolutely zero flex but the seat post still flexed a great deal, you would ride that bike and still think it was flexy. You cannot separate the seat post from the frame outside of a laboratory.
I doubt that this is true. I have never seen a new frame set sold with a seat post. If the seat post was flexy it might contribute to a "softer ride" but it would not cause the chainwheels to rub on the fd or cause the chain to jump cogs. Flexy frames generally show up when hammering out of the saddle.