I'm not understanding something here.
When we say that ancient egyptians were able to predict eclipses, the answer is that through centuries of observation you can discern patterns. You dismiss any relationship between these observations and an actual model of the universe, you say that through observation alone you can predict these things.
If that pattern is described by an equation, why does it become illegitimate? Why is it less legitimate to predict the sunrise from such an equation or pattern of observation than it is to predict eclipses? This doesn't say anything about flat earth or round earth, just that "at latitude X and longitude Y, we predict sunrise at time HH:MM" - and that prediction is correct day in and day out for thousands of years.
I don't think you question eclipse predictions, or timing of the full moon, or any other observations that have been made for thousands of years - and when it is so easy to see a mistake if one occurs, why can't we accept an equation instead of observing the timing of a sunrise?
If you ever find the prediction to be in error you have your "a-ha" moment.