The second layer of satire that most viewers missed?

I think we all understand that this is primarily a political satire and it's quite obvious what it makes fun of - corrupt political system, media and people stupid enough to buy into what they tell them. But what if there's another dimension to it, a bit more subtle than the rest of the movie which is mostly in-your-face? What I think is that on a deeper level, this movie actually makes fun of conspiracy theories, which is ironic considering that a large portion of the fans are conspiracy theorists who take the movie almost literally.

The entire plot sounds like something that came out of Mel Gibson's imagination in Conspiracy Theory.

Take for example the making of Albanian girl with a cat. They make everything in front of a green screen with a fake cat and professional actress. They didn't even bother to use practical effects, shoot on a real location, or anything like that. Didn't we all hear this type of conspiracy theory at least once, where someone claims that the whole event (moon landing, 9/11...) is faked based on supposed mistakes, such as "look, they forgot to add stars when they faked moon landing in a film studio."

Then you have the Old Shoe song that was faked for propaganda purposes and claimed to be made in 1930s, USA going to war only so that the president could win an election, Brean says that JFK actually died in a car crash, and so on. All of this is not how conspiracies are conducted in the real world, but it sounds exactly like something you'd hear from a bad conspiracy theorist on a radio show in the 90s. The whole story is just intentionally made to be as absurd as possible, possibly as a satire on conspiracy theories.