What happens when the concept of map projection distortion meets religious fervor?
Exactly. These venerable sites are not even close to lying on a single great circle route. Furthermore, the claim about sunset at the summer solstice is geographically impossible. Moderns, flattering themselves as scientific, usually are completely innocent of any grasp of seasonal earth orientation.
That's a quite unfair observation to make. You're only using facts that support your position. I'll bet you never clapped in Peter Pan to save Tinkerbelle either.