Every statement you made was false.
Prior to 1953, Iran was a constitutional monarchy. Mohammad Mosaddegh was not appointed by the Shah, as you claim, but
elected by the Iranian parliament. The Shah also wasn't nearly as independent as you claim; his deference to the United States was part of what led to his eventual ouster. He also wasn't supportive of communism. The communist party of Iran (which supported the nationalization of Iran's oil industry during Mosaddegh's time as Prime Minister) was banned by the Shah. The Soviets even tried to assassinate him, according to defectors.
As for the sources relating to Operation Ajax being from the CIA, well yeah. It was a CIA operation, who else should have known about it? But it didn't occur in a vacuum. The UK wanted Mosaddegh overthrown (remember who owned the oil industry that Mosaddegh wanted to nationalize). So your version of the story is what? That the Shah staged a coup coincidentally at the same time that the UK wanted the US to do the same thing, and the CIA then faked documents taking credit for it? Yeah, sure.