hounddawg asks an intelligent question:
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once the case enters the die what does press stiffness have to do with anything? The case is free floating in the holder, or at least has several thousandths free movement in any given direction.
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Yes, the case can move in any direction horizontally, and typically does in the shell holder or with the shell holder in the ram's top.
But its the long axis of the case relative to the die's long axis that is the issue.
Depending on the type and amount of lube on a clean fired case as well as the taper of a full length sizing die's body and the diameter differences between the die and case, as long as there's some clearance between the bottom of the die and the top of the shell holder when the ram's all the way up, the amount of spring the press has will cause a dimensional problem with sized bottleneck cases. Sized case headspace, the distance from the case head to the reference point on the case shoulder will vary. Sometimes as much as .004" to .005" depending on all those factors mentioned.
If the die's not set down enough in the press, some cases so sized may not chamber easily. Others may have their shoudlers set back too far which leads to case head separation at the pressure ring. And best accuracy as well as case life happens when the fired case shoulder's set back no more than .002" and there's no more than a .002" spread across all the sized cases' headspace.
Therefore, a press with the least amount of "spring" typically produces more consistant size case dimensions for bottleneck cases.