(not even in his private correspondence, nor even when death was coming)
Even if he were part of a global(!!
) conspiracy, I'm not sure why the shape of the earth would have been found in his private correspondence, and especially not towards the end of his life. He was pretty singularly focused on other matters in both respects.
But you do bring up an important - if broader - point.
And you don't have to go that far back into history for it. I would have asked, why didn't the Soviets, when their space program was in clear decline and their army revealing itself to be in shambles in the late 80's, have gained an easy PR victory against the democracies of the West? True, this would have been a fleeting victory, but that's hindsight. All the way up until 1989 most in the Kremlin did not realize what would happen a mere two years later!
Furthermore, look at great power politics today. Here's a few examples that could imply one nation, if they were party to a grand conspiracy, would have a motive to reveal "the truth" to the world:
China has a keen interest in showing the world that it is a true peer to the US, to the point of making itself out to be a superpower (which it is not, by accepted historical standards). But it is a geopolitical rival to the US on the world stage, playing the part of disruptor to the unipolar status quo and hierarchy. Like most autocracies before it, it often chooses to point out perceived moral flaws in its rivals to distract from its own repressiveness (both to domestic and world audiences), or to gain PR points in the game of great power politics. And it often seeks to show off its military and technological prowess. One way it could kill multiple birds with one stone would be to reveal that the earth is flat, and to demonstrate this fact to the world. In short, China would be saying that the US, by lying to the world about such a fundamental fact, can't be trusted as de facto world order hegemon. By contrast, China will play it straight with you. Be our ally.
Dozens of Arabic nations, as well as Muslim Iran and statelets like Hamas' Gaza and the PLA routinely attempt to show up Israel diplomatically or geopolitically. Israel is by far the most technologically advanced nation in the Mideast (and so if any of them know the true shape of the earth, they should), and they are presented as irredeemable liars and conspirators by a great many voices and groups in the Muslim world. It would be keeping exactly in line with many routine practices for, say, Iran, to tell the world Israel was the originator and gatekeeper of a massive conspiracy to hide the shape of the earth. And then Iran could demonstrate that the world is flat in order to embarrass Israel and show the world they cannot be trusted. If they lied about the very shape of the world, how can we believe anything the Jews say?
An important point regarding this second example: if the base reason for the conspiracy has to do with dominance in space, Iran is very new to having any sort of space program, and has little to lose in this arena (e.g., if their space program is indeed a farce like NASA, Iran has no levers of geopolitical power founded on its own space program, so it loses nothing by revealing the earth is flat).
Repeat this concept with Russia, using the truth as a weapon against the EU or US.
Hilariously, you could also repeat this concept with Ukraine, using the truth as a weapon against Russia, even, to further garner the support of the civilian populations of the EU and US!
Repeat it with N. Korea against S. Korea/Japan/US. (I admit this one is more dubious given N. Korea's general silence on the world stage).
Heck, you could even repeat it with Armenia and Azerbaijan. Although less well known and much smaller countries than the examples above, in late 2020 Azerbaijan kicked Armenia's ass in the Second Nagorno-Karabakh War with impressive technical sophistication. It could be called the "first robotic war", as it was the first to be won primarily by robotic systems. Azerbaijan could immediately make itself and its ongoing conflict with its rival much more famous by demonstrating to world audiences that the world is flat. Of course, this example does lack any realistic argument that Armenia has been some sort of key conspiracist for the shape of the world which seems dubious (unlike the US, Israel, Japan, or the EU, as in the other examples). On the other hand, Azerbaijan's purpose could be to enlist popular support from major rich nations like those in the EU and the US. I guess this one is a bit far fetched. Still...
All this implies to me that there is no such grand conspiracy.
But these are also very complicated situations, and I can believe that others could honestly think all the above parties have reasons to keep silent even so.