The post is indeed not a great circle.
Redraw the path nearest to the red line but avoid the areas marked in green and you have your answer. Arizona is at a pretty high elevation and the rockies stretch from new mexico through colorado onwards, so they probably fly around the other way under the Rockies.
Edit: i think his question is still disingenuous
He's not asking because he understands great circles or is suddenly curious about planes. He's "just asking questions", specifically on a public forum to drum up disinformation and anti-science rhetoric. He's essentially giving a shout-out to his conservative skeptic gang that if powerful rich people question science and common knowledge, they should keep doing it too