How does one go about solving for cognitive dissonance? It can be hard, but it's doable. Take a step back and address the claim head on, like yes or no and why it is or isn't true. Then if you want to make a different argument that carries with it implication that the impact of the previous claim is not that important, that's fine.

Simply just sidestepping the claim and presenting a new claim masquerading as a response to the original claim is how most arguments seem to go, and it never gets anywhere.