As my last note on the subject, I'm convinced more than ever that we're not using the same vocabulary. We must be seeing the same things, but they mean something different to each of us. I'll once again back up until I think we're on common ground...

All I can point out is, again, that the article you posted is not talking about the position of mercury in the sky. It's talking about a particular observation mercury is seen to do when the pattern of its motion on the sky repeats itself. Einstein's version of gravity is a little different, and so on a high level "basic gravity simulator," or otherwise simple model, the bodies closer to the center would move a little differently.

Since the Flat Earth theory also has the planets moving around the sun, with Mercury at the closest position, I don't find it odd that a different version of the phenomena that keeps the planets moving around the sun could cause the most interior body to move a little differently in its repeating patterns in the sky.

This is a basic modification to a high level and theoretical system. But we are just talking about basic orbit types on a basic and high level model, and slight adjustments to make some interior planet move faster or slower based on your imagined central pulling phenomena. It is an imagined explanation.

To prove the model there would need to be a prediction of a body in the sky and expression of the celestial mechanics. If the article you had posted claimed that Einstein predicted the position of Mercury in the sky this would be a very different conversation. He did not do that, and the articles surrounding this event admit as such.

But then once we have that, how do we tell where they will be in our sky at any given place on the Earth? In that NOAA spreadsheet you have mentioned, I would like to draw your attention to cells B3 and B4.

These are the observer's latitude and longitude. The locations of the bodies is presented given the observer's lat/long under the assumption that the Earth is a globe. Take a look at cell W2:

=DEGREES(ACOS(COS(RADIANS(90.833))/(COS(RADIANS($B$3))*COS(RADIANS(T2)))-TAN(RADIANS($B$3))*TAN(RADIANS(T2))))

Notice that the latitude is going into some trig there. That isn't projecting the sun onto a flat earth. That's RE math there.

Why do you think it has anything to do with a Round Earth? Cell W2 is the sunset column. The time of sunset must have some sort of relationship to do with your longitude, even in the Flat Earth model. Using trig manipulations to find the relationship is entirely possible.

Edit: One last note. You seem really fascinated with the N-body problem. We have no analytical solution for this. Meaning, there does not exist a mathematical formula to solve this in general. All that means is we have to do it with computer simulations instead. Not really a flaw in RE in any way.

The impossibility for an analytical solution of the Three Body Problem means that we can't create formulas to turn the positions and movements of bodies into equations that will predict future occurrences under the heliocentric model. It means that it is impossible to create a program like Stellarium that predicts things under a Heliocentric or Round Earth model. The idea that such programs are possible is the crux of many arguments that the Round Earth model is true.

If such programs are impossible, and the prediction of bodies is really made on patterns and trends from past occurrences then it means that the Heliocentric Theory is much weaker.

It is more of a blow and punch to the nose. It shows that the model is much closer to a hypothesis than its supporters believe.

The inability to predict suggests that it is either the science or the model that is incorrect, or both. The inability to predict the positions of bodies does not lend support to the heliocentric model, and only subtracts from it. It detracts from the reputation of classical mechanics, mathematics, astronomy, et all.