In the RE model the earth is orbiting the sun and the moon is orbiting the earth. With all are knowledge of gravity we are unable to calculate this orbit. Mathematically it does not work. Google the three body problem.
There is no analytical solution but there are numerical solutions where you take the initial conditions, break the problem down into several 2 body problems and in time steps solve those. Those do perfectly fine as an approximation - good enough to accurately predict the positions of the bodies for all practical purposes. The model is good enough that when the solar eclipse happened the path could be predicted to block level.
Neptune was discovered because the wobbles observed in the orbit of Uranus indicated there must be another body pulling on it.
The model of gravity works very well in most circumstances. Einstein showed Newton wasn't right about certain things but for most practical purposes it works.
I'd say the failure to make an analytical model is more a failure of mathematics, not science.
Many people have come here saying that the sinking ship effect has come as a result of the earth being round yet, it has been demonstrated many times, that the sinking ship effect can also be caused by refraction or chaotic atmospheric conditions.
That certainly does affect results but do you know what I've never seen? In all the pictures you see where a FE person says "aha! you shouldn't be about to see 'x', but you can". I have never seen the whole of 'x'. What I mean is, I've seen videos where people say you shouldn't be able to see the top of a mountain, say, but you can. So they conclude that the earth is flat. But if the earth is flat then where's the rest of the mountain? Yes, atmospheric effects do mean results vary but on the right day you'd surely be able to see all of the mountain or whatever. But you never do. Although yes, there was that experiment with mirrors, which brings me to...
The bishop experiment is one which had a similar experiment done with mirrors which, a very firm 100% round earther admired that it was evidence that supported a flat earth.
The Bishop experiment is really poorly documented to the point where I discount it completely.
But yes, Bobby did agree that the mirror experiment was possible evidence and I'd agree with him.
BUT, he attempted to reproduce the experiment and couldn't reproduce the result.
The reason there are so many different FE models is the mentality is "check things out for yourself". Now, that in itself is not a bad principle.
But there are some things which are quite difficult to check out for yourself. I don't have the means to do the Cavendish experiment, for example, which is evidence for an attractive force between objects. And maybe I'm not very good at experiments and my errors will lead me to wrong conclusions.
The whole basis of science is that people do experiments, document their methods well and then
publish the results. That way other people can scrutinise their results, repeat their experiments and that can help push discovery on. Someone else might find an error in the method, or maybe other people get the same result and that gives confidence in the result. That's how progress is made. The idea that everyone should do their own tests and come to their own conclusions is why there is such a mess of competing and inconsistent models.
Given the wealth of evidence for the globe earth I don't understand how people can do experiments, get a result which indicates a flat earth and think they've discovered something so revolutionary and which has such massive implications in terms of the myriad of conspiracies which must be at work...rather than thinking "maybe I suck at doing experiments" or "maybe I don't understand this stuff as well as I think I do". A lot of the FE arguments I've seen are arguments from ignorance or incredulity which is not a very sensible way of forming a world view. We shouldn't blindly believe everything we've been told either of course but there is a middle ground between that and thinking that each person can form their own model based on their own observations. The inconsistencies in your results should demonstrate that. The earth is the shape it is, the sun is the distance it is and so on.