> If QED says light travels in the path that minimizes the time of travel

It doesn't say that. Classical optics says that if diffraction is negligible.

> shouldn't all the light from an object become part of the mirage image in situations involving mirages?

No. Why would it?

> (I.e., why would there be a "real" image as well since that light takes longer to arrive at the observer.)

The direct image will take less time, but it's on a completely different path so that comparison doesn't matter. The shortest path statement only applies to adjacent paths.