No one in the government is actively lying about the Earth's shape (as far as we know). They genuinely believe it to be round. The lie is the act of having advanced space travel capabilities. Back in the late 1940s, the 50s and the 60s, governments were just beginning to understand the properties of mass propaganda and how it can shape the reality of their people.
The USSR always existed on the fundamental idea that they were working towards some glorious, super-industrial future that never came to fruition. What better motivation than to tell their people and their enemies, that they have sent man-made objects into space? Amazing, finally their glorious technological future will come to pass! Except it didn't. They never sent anything into space. They lied to their own people. However, the West couldn't just sit idly by because their own people believed the lie too. "Look at the USSR!" they said, "they're so advanced!" The West couldn't allow such a thought to start purveying their own populace. Something had to be done.
The West couldn't simply say the USSR is lying. The USSR would continue to lie and people would start to ask "well why aren't we sending things to space?" Therefore it was decided that the West would lie as well. "Look, we've sent stuff too!" Then it became a big game of who can con their own people into the biggest lie. It became a game of "let's get Kubrick to televise our lies on national television". The USSR couldn't compete with Hollywood. They lost the lying game.
The Earth not being round is just a part of the big lie, it's not even an intentional lie. If the US thought the Earth was flat when it faked the moon landing, the Earth would appear flat in the movie. It's just an artistic backdrop picture. Space travel still is, and will continue to be, a silly lie, which is why NASA will fade into irrelevance, why SpaceX only exists because of government contracts, and why every company who ever uttered the phrase "space tourism" is doomed to failure. Eventually the government, and schools, will tell us all that there's nothing interesting in space, don't bother thinking about it, and they'll hope faking the moon landing is no big deal when it's declassified in a century or so.