Of course you are right about Galileo.
We are thinking about the "apparent" motion of the sun around the Earth becuase it is a bit easier to picture the Sun moving around the Earth than the Earth moving around the Sun.
But the answer would come out the same if we did this problem the correct way as you suggest.
We said the sun would travel eight minutes while the light we see if travelling to us. Looking at things your way, the Earth
would revolve eight minutes worth of revolution while the light we see is travelling to us. Thus you will still be pointing in the wrong direction.
Either way the math is the same, and the answer is "correct" given the caveats I gave above.
The ancients viewed Space as a "celestial sphere". The stars were painted on this sphere with the Earth at the center. The Sphere rotated once every day (with all the stars etc.) The sun and the planets moved very slowly about the celestial sphere.
Of course the ancients used this model becuase it matched with what they observed. This model was simple, yet it explained and predicted what they observed in the sky.
This celestial sphere is still used by astronomers today, even though we now understand the new model. Astronomers still talk about the celestial sphere, and measure motion in the sky relative to the "celestial equator" and the ecliptic (the apparent path of the sun through the celestial sphere over the course of a year).
Modern astronomers do this becuase the motion of the celestial sphere much easier to think and talk about than the simultaneous multiple motions of the rotation of the Earth, its orbit aound the sun, and the Earths precession etc. As long as we understand and keep in mind that this is a simplification, it is a helpful shortcut.
For questions of celestial geometry it comes up with the right answers and makes life a little easier.
When I was teaching astronomy in high school I used a globe that was a model of the celestial sphere. It was transparent and had the stars painted on the inside. There was a small model of the Earth in the center and a little sun that moved along the ecliptic. This model is invaluable for explaining the motions in the sky. It is also instructive to compare these "apparent" motions to the "real" galilean model.
For more information see...