It's a plane, not a planet!
Now I have only begun to study this topic, and largely, I'm going by instinct.
One scientific proof that you'll immediately scorn, I presume, is the sideways look at a moving ship. It's in Rowbotham. He stretched a rope across two poles near a shore, and watched a ship out at sea numerous miles away, moving parallel to shore. He says the ship didn't change in altitude, despite the fact that over the curvature that should have been represented by the length of the rope, projected out to where the ship was, would have meant that the ship should rise and fall again.
I think, in general, that it is more beautiful for the world to be flat. Therefore it is probably true.
I find that there is a tremendous difficulty in clarifying the matter one way or the other. Many discussions are held, and people seem to feel that if the matter isn't clear, then spherical-earth wins. But if it isn't clear, it isn't clear. A bad argument is only that.
It may be that we're not given to know.
There are too many problems with the science of spherical-earth theory. Why shouldn't gravity within particles cancel out gravity within equal-and-opposite particles? I understand the concept that gravity keeps us all together, but why shouldn't a cannonball shot upwards come down in a different spot? It just seems like it really should. It's amazing that airplanes don't have funny problems with the multiple movements of the earth. But again that's gravity for you.
And I am also that beast, the Biblical literalist. Until recently I didn't think anything of the flat earth, but what the hell? Why can't it be flat? Maybe it is. I don't want to discuss the Bible much because it's not the place, and any discussion would sound like Bible-thumping, something I always hated before I took it seriously. So no panic. But the Bible is incredibly prescient when you know how to read it. There is no other set of books that could come anywhere near to it, if one wanted to argue that it's a matter of twisting this and special-pleading that.
Rowbotham and Scott claim that concentrated Moonlight doesn't add up to Sunlight in its effects. But the Moon is supposedly reflected Sunlight. There are other aspects of Moonlight that they mention, for example, it is visible during some lunar eclipses, suggesting that the Moon emits its own light. I always wonder if people are joking around here when they mention Moonlight's effects, but, I do know that people who work with flower essences believe that Moonlight and Sunlight are two different entities.
The science of spherical earth theory was clearly brought forward as a theory, but now it is taught as fact with no counter. This matches a pattern I look for: an idea taught without the contrary idea being permitted mention. I have long known to suspect that pattern as prima facie evidence of error. Not a proof, but noteworthy.
I'm supposed to be 'used' to rotating and gyrating at huge speeds, but I don't buy it. Even going steadily in the car, I feel the motion. So the theory then has to be that the earth's movement is unbelievably perfect. Okay, that's the theory. I still don't buy it.
In many cases people are trying to reason back to earth by something they think they observe about something going on outside/above it. These proofs don't convince me of much, because they typically assume what is not proven: this is begging the question.
In sum, I think that spherical earth theory is prone to imaginary systems that are less necessary than their adherents believe.
My two cents!