Why are people with full suspension bikes concerned about 'Stiffness' added by thru axles? Why are people with full suspension bikes with several inches of travel be concerned about 'Stiffness' when discussing the advantages of thru axles. I would think this is more of a concern for rigid bikes which don't have suspension. Isn't the advantage that you're just less likely to snap the axle when riding down a gnarly trail not really how 'stiff' the axle is or does this have some other purpose?
Thru axles not only keep the wheel in place, they also strongly join the right and left dropouts together. This structurally reinforces the component either being it a fork or a rigid/suspended rear triangle.

This simply makes the bike able to handle greater forces before failure, because it distributes forces between both sides of the structure. It also avoids twisting of the parts. For example, if one arm of a fork twists, it means it may loose alignment relative to the other, thus meaning the wheel is not pointing in the direction the handlebar tells the rider, this leads to a bad feedback and imprecise ride feel. A thru axle keeps both sides of the fork aligned, eliminating (or reducing) this issue. It also keeps the lower ends (bottles) better aligned with the upper parts (stanchions), allowing for a smoother operation and longer  life of seals and bushings.

For suspended rear triangles it also makes the structure stiffer, this allows better distribution of forces through the stays, so those forces reach the shock absorber with better alignment, also allowing for smoother operation and longer component life. It also keeps the pivots aligned, thus giving them a longer life for both bearing type or bushing type pivots.

The bottom line is that a stiffer structure is more efficient and gives better feedback to the rider, allowing for a more confident and precise ride. It also allows a lighter bike to handle greater forces.