You can grill me as it is good to ask questions, though I may not have the answers.

If one accepts the scientific methodology employed as being reliable and valid then I have no problem accepting it as fact. However carbon dating is subject to carbon contamination from other organic sources in the environment thus making carbon dating results suspect. Same thing with evolution. I believe in micro-evolution but I don't believe in macro-evolution due to the absence of intermediate or transitional forms. We could debate this till the cows come home but you ask the pertinent question: "What could someone gain from promoting a spherical Earth and keeping a flat one secret for the past 500 years?"
That question goes to the heart of the matter so let's deal with that first. It is not just the flat earth but what exists above the earth. Gen 1:6-7 states that there is firmament above the earth. The flat earth model proposes that there is a dome above the earth which separates the waters above from the waters below. Is the "firmament" described in Genesis 1:6, heaven or a solid transparent dome? "The heavens declare the glory of God; And the firmament shows His handiwork"(Ps 19:1). This verse references the heavens and the firmament as two separate things so since it is not heaven, can the firmament be descriptive of an actual dome? I know that sounds crazy but you can do your own research. Incidentally, Ps 19:1 verse just happens to be on Wernher Von Braun's tombstone. Coincidence? You may want to start your research with the Operation Fishbowl high altitude missile tests in 1962. The name is curious and lends itself to speculation as to what exists above the earth. Do we inhabit a "fishbowl" domed earth? The government has publically stated the reason for conducting such tests but were there ulterior motives not made public? For the sake of argument lets suppose the launching of these missiles exploded against a dome thus confirming the existence of such a thing. An impenetrable dome would make the moon landing a hoax of massive proportion, it would negate satellite technology and sending probes into space exploration a part of that deception. It is speculated that satellite GPS does not exist and radio signals are bounced off the dome to acquire location coordinates on the earth. If indeed so, why such elaborate and expensive deceptions to conceal the truth? My answer is that it all relates back to Matt 24:24 where it states that even the elect will be deceived. If we live on a domed earth, there can be no such thing as space travel and no aliens traveling in UFOs. Aliens are nothing more than demonic entities who will make their appearance in the last days to set up the coming deception. Belief in the "existence" of aliens would not have been possible today if mankind still believed in a fixed flat earth with a dome instead of a round earth revolving around the sun. That world view had to change and the change occurred a few hundred years ago. Ever wonder why the Catholic Church uses a telescope in Arizona with an instrument called Lucifer? So the answer to your question is that a conspiracy did indeed have to take place - a Luciferian one - where the biblical world view had to be supplanted by a scientific one centuries ago, in order to arrive at the notion today that mankind is not alone in this universe and we are being visited by visitors from other planets or galaxies. In order to be accepted and even welcomed by mankind, these demonic entities who travel interdimensionally know that they have to disguise themselves and pose as [benevolent] aliens who travel interstellary. Therein lies the deception as Satan who is the father of lies has spared no expense and effort in foisting and fostering this lie for the past few centuries in order to set up his end-time deception.
Ever wonder why Admiral Byrd who was tasked with exploring the South Pole (Operation High Jump) - again a peculiar name - has stated that "there exists an area as big as the United States that's never been seen by a human being, and that's beyond the pole on the other side of the South Pole from middle America." What? How can there be land beyond the South Pole on a global earth? It could be possible on a flat earth however where the South Pole is situated all along the perimeter edge of a circular earth with the North Pole occupying the center of a circular earth as flat earth maps depict. Either Byrd was out of his mind, not telling the truth, or accurately reporting the results of his expedition. Youtube has his interview, just search on it type in "more land beyond the antarctic proof." One must ask why is Antarctica protected air-space and except for guided tours generally off-limits to civilians?
As far as sailing around the world goes, in the past sailors had to sail around the Cape Horn to travel from the Atlantic to the Pacific. Now they can opt to travel through the Panama Canal. On a flat earth it makes no difference. On a flat earth, picture if you will the continents surround by the oceans. On a flat earth one still needs to circle the Cape Horn or travel through the Panama Canal as one sails east to west or west to east. The only difference is that one is not circumnavigating a globe but sailing in a circular route along the perimeter of a circular earth to travel around the continents.