I hate to think of the implications of that. Jimmy Carter was our president when the Shah was run out of Iran in 1979. I don't actually know much about why we favored the Shah and in what way we were involved over recent decades.
It goes back farther than that. While the BBC timeline you linked to is accurate, it is very incomplete.
To begin with, the Shah was hated by lots of Iranians for multiple reasons. We deposed the elected leader and replaced him with a king because he was very Western oriented. Mossadeq was inclined to align with the Soviet Union and nationalize the oil fields. He would also have preserved traditional iranian culture and mores. The Shah was giving away iranian wealth to Western interests and forcibly secularizing Iran. So he was hated by everyone from conservative Muslims to the modern intelligentsia. His grip on the Peacock Throne was tenuous to say the least. He absolutely needed support from us, which means he's a western puppet to proud Iranians. And the way we kept him in power was brutal. His secret police force, SAVVAK, was nearly a division of the CIA. The torture and disappearing of dissidents, spying and such was fearsome.
Then along came the OPEC thing in 1973. That majorly roiled all the waters in the oil producing and consuming world. Caught between those two poles, the Shah was in a world of hurt. SAVVAK was in high "crush all dissent" gear and really ugly. Then Jimmy "The Christian" Carter got elected president. He was way too ethical to allow our government to be responsible for the kind of horrors we were committing in Iran. This was while future president GHW Bush was high muckety muck at the CIA, and keeping control of the Iranian oil fields was a top priority. But without the president's support he failed, the Shah was overthrown, and Khomenei took power.
That's why the USA embassy in Teheran was such a big deal to the Revolutionary Guard. It really was CIA headquarters in Iran, source of all evil in Iran. A nest of enemy spies and agents and people guilty of all sorts of crimes against the people and government of Iran.
Eventually, Bush used Saddam Hussein to launch a horrible invasion of Iran. But that's a whole 'nother episode to the story.
Lots of people either don't know all this, or even deny it happened. But that doesn't matter to the Iranians who do. And they are the ones we're dealing with.
Tom