Use of axis of symmetry Does an axis of symmetry determine chiralty?

Is axis of symmetry considered a true symmetry?

From the above two links, I read that a compound having axis /alternating axis of symmetry is NOT necessarily achiral. For example, trans-1,3-dimethyl cyclopentane has an axis of symmetry but it is still chiral and hence optically active.

My doubt is, then why would we look for alternating axis of symmetry in the first place? Where/How is it useful?
Alternating axes are not axes, much like a Guinea pig is not a pig.

Having axes of symmetry is irrelevant to chirality.

Having alternating axes is a necessary and sufficient condition of non-chirality, that's why we look for them.

So it goes.