It was never "true" that the earth was flat. Obviously the earth was round before people figured out that it was round, but there were no experiments which "confirmed" that the earth was flat. The earth was demonstrated to be round by experiment in the third century BC, and we have no evidence of any experiments predating that which indicated otherwise. The roundness of the earth was not discovered by the "common sense" of ships vanishing over the horizon.

Experiments showed that the earth was round, to within a small margin of error. Further discoveries showed that, within that margin of error, it was not perfectly round. At each step, experiments advanced our knowledge, rather than misleading us. There is actually a specific coined term for the position you're arguing, using the same argument, and this is it.

Common sense has lost out quite a lot in science. Einstein himself, for instance, criticized quantum mechanical models, arguing that if they were true, we would expect nonsensical things like quantum entanglement, so we should be pretty sure that the models were wrong. Unfortunately, reality came back with a resounding "screw you Einstein, quantum entanglement is totally a thing."

Math, on the other hand, has a much better track record. We do not have a history of mathematically proving that something should never happen, for instance, and reality comes back and says "tough, I'm doing it anyway." We have a track record of working out mathematical formulas that work when you apply them to abstract numbers, and then when we use mathematical models to try and make predictions about the real world, they actually work. Do you know how black holes were discovered? First, people did the math, based on our models of physical law, and found that if you plug the numbers in, if you pack enough matter together, black holes are what you get. It wasn't until years later that we made the observations that showed that black holes really do exist.

This is why scientists use mathematical models of reality, not just "common sense." It works. If it were easier than common sense explanation, we would have been doing this thousands of years ago, but it's not. And so the people who have computers, satellites, phones and airplanes, are us, the people living in a civilization founded on the fruits of science, and not the human race preceding the advent of science. It's hard to argue with consistent, reliable results.