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Do thru-axles not have higher clamping forces? Stiffness can also be noticed between QRs, say between a superlight PX ti skewer vs a Shimano closed cam skewer. Where the former might lead to the brake pads rubbing when climbing and the latter will not. @mdcc_tester might be able to explain better than me
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I think the point is that in a traditionally spoke-d wheel there is relatively little rim movement (deflection) side to side as it is pulled by each pair of spokes. This means that you notice the issue at the hub more, which is why the different skewers make a difference there.
AFAIK the issue with a carbon wheel is that it fundamentally isn't as laterally stiff as a spoke-d wheel so they will always be flexy
You are right.
But the through axle becomes part of the fork assembly and doesn't change the physical properties of the wheel.
In order for the wheel to become stiffer you would have to increase the flange diameter of the hub or use a different material altogether.