Well, we can do another simple experiment to test the same hypothesis. I call it "The Human Sundial Experiment".
Here's how it works:
At any time during the day, when the sun is unobscured by cloud and shadows are cast, we can see that the Sun appears to move across the sky at roughly four times its own diameter every eight minutes. The shadows it casts "move" across the ground at the same angular velocity. That's how a sundial works - a sundial shows how the Sun appears to move across the sky at a steady rate that matches the rotation of the Earth.
If you stand near a post, building or other structure, theres a spot you can be in where Sun can be 'hidden' behind that object, so that you can't see it. Lets pick a telegraph or power pole (you could use a tree or a shed, or whatever, if you prefer, as long as it's tall enough and thin enough that there's somewhere to stand where the Sun is only just hidden behind the pole).
Pick a spot to stand, facing the Sun, far enough back so that the Sun is only just blocked by the pole - so that if you move even the tiniest bit to either side, you will see the edge of the Sun.
Now, the ground at your feet is illuminated on either side by sunlight that left the Sun eight and a half minutes ago. The shadow is pointing directly at where the Sun was when that light left the Sun - if it wasn't, the shadow wouldn't be where it is*.
But the Sun has moved since that light left it. It is eight and a half minutes, or four solar diameters, further along its apparent path across the sky than it was when it sent out that light.
So, if we see the Sun instantly, with no delay, then we should see it, four 'pole widths' to one side of the pole, while we are standing in the shadow; And when we stand so that the Sun is exactly blocked by the pole, we should be in sunlight, with the shadow falling off to one side, four times it's own width counter-clockwise from where we are standing.
That's an unavoidable result, IF we see the Sun instantly, but see the light reflect off the ground only after that eight and a half minute delay. The spot where the Sun is completely obscured from view should, according to your hypothesis, be outside the shadow cast by the post.
This is an observation that is directly implied by your hypothesis. If you are right, then the above is exactly what we must see. It's also not what we actually observe, if and when we do the experiment. Don't take my word for it - do the experiment for yourself. Anyone can, on any sunny day.
*The Human Sundial Experiment is functionally much the same as the Sunrise Experiment; In the latter, we used the Earth itself to cast the shadow, but in this new experiment, we use something that's small enough so that the Sun is only just obscured from our vantage point. The benefit of this is that a telegraph pole doesn't have an atmosphere to scatter light, so we aren't at risk of being confused by the pre-dawn brightness of the sky.