None of the flaws I've said are specific to academia dealing with FET. In the modern era, the response to a breathtaking new discovery is to append, not to replace.
Agreed. Because if you have 1,000 observations that can be explained by Theory X, and then you have a new observation that doesn't seem to fit, you can either A) see if there might be something you missed in Theory X that, if corrected, can now explain 1,001 observations, or B) create Theory Y that can re-explain the first 1,000 observations as well as the new one.
Science will always start with modifying existing theories because it's a smaller lift than rebuilding everything currently known. If that doesn't work, then it is forced to build a new theory.
Let's walk through your example. Again, I don't think it will have the result you're claiming, but that's not the point. I perform this experiment, I even manage to get it published, scientists all around the world see and even believe the data on this new bedford level-esque experiment. Even if it is conceded that that would happen, the community at large might:
1. Brush it aside and ignore it.
2. Pay attention to it, analyze it, shoehorn in some explanation of light interacting with a dark matter temperature inversion or some such and use it to add more and more to the existing model, test a few dozen hypotheses until one happens to be in line with something by force of numbers, take that as the accepted explanation.
3. Start questioning whether the Earth was round.
The goal here is to reach 3, I am claiming they will do 2. Are there any objections so far? And if not, how might we reach 3?
What you're outlining in #2 is revealing what I think is the underlying fallacy in your understanding of how the scientific process works.
You said, "test a few dozen hypotheses until one happens to be in line with something by force of numbers, take that as the accepted explanation." Yes, you start testing your hypotheses, and the ones that fail, get discarded. That leaves you with the ones that didn't fail, and as newer ideas and better equipment comes around each year, you test them until they fail and get discarded. All the while, you're probably coming up with some new hypotheses, testing them, and discarding some. When you hit a hypothesis that keeps passing every test you throw at it, it starts gaining more weight.
It's not a game of tweaking esoteric equations to make things just damn fit. It's predictions each hypothesis generates that gets tested.
As an example: The biggest question in physics is how to reconcile general relativity with quantum physics. We know they both can't be right, but they are both amazingly good at predicting what our experiments will find. String theory actually solved this conundrum for us - BUT, string theory doesn't generate any meaningful predictions. The math definitely lines up, but there's no way for us to disprove the hypothesis. So now, string theory is losing its luster and researchers are looking at other avenues. So it's not about the math just working and then scientists say, "Okay, that one works. Let's just go with that."
If you did solid research that showed an apparent lack of curvature over a 5-mile stretch, scientists would say, "That doesn't fit with all the other experiments I've seen that show the opposite. What's going on here?" They might say, "It's refraction," but you would already have thought of the top ten objections because you would have done research before you designed the experiment in order to account for something as obvious as refraction. Another researcher would say, "It's due to phenomenon ABC," and you'd figure out how to account for that and do the experiment again. In the meantime, other scientists would be wondering what's going on and would start trying to reproduce your experiment for themselves. If it keeps holding up, they'll start trying to disprove it by looking at some of its immediately testable predictions and setting up new experiments.
If it holds up under everything they throw at it, then they'll start trying to understand if there's a new phenomenon happening. They'll keep backing it up until they reach a point where math and hypotheses and observations all start to mesh again. That may very well mean backing all the way up to a flat Earth. That would be a very long way to back up, however, because as I mentioned earlier, physics itself would have to be gutted and rebuilt.
So it's not a single step from doing the experiment to the world accepting a flat Earth, which is why it hasn't happened yet. It takes an experiment done with extreme rigor that shows a lack of curvature, then legions of researchers trying to replicate, disprove, and enhance your experiment, then a process of tearing down every bit of conflicting science that is less certain than your evidence, and a new theory with new predictions that starts getting tested. That's a stunning amount of work that would need to be done just to break down the current paradigm, let alone build a new one that leads to the conclusion you're hoping for. It would likely take a decade at minimum since so many sciences would be affected.
With all that said, you can see why an experiment that would set all these wheels in motion absolutely has to be impeccable. Fortunately, there are a lot of simple curvature experiments that can be done inexpensively and with a great deal of precision.