Is everybody ready for the US flavor of sharia law?

I don't think their comment was pro-Iranian-post-revolutionary-government. The Shah was awful, was installed by the CIA, and did kill political opponents. US-backed governments don't typically fall to revolutions, so a lot of people must have been upset to have enough of them to manage that. It's generally accepted that what came after the revolution was worse, but it's not just nutters, Stalinists and tankies that recognise it was also bad before, and got that way because the British wanted oil.

As far as I'm aware (which is a bit more than average as I'm British with an Iranian grandparent), both of you posted correct things. If the Shah hadn't started killing anyone who disagreed with him, it would have been harder for the religious extremists to kill the rest. It's not like you can ever assassinate all your political opponents as everyone knows other people, and those people don't like their friends and family being murdered.

Oh, Iranian history is a special interest of mine, if you express the slightest interest in it...

The roots of it go back to the 1800's, when the Qajar dynasty were trying to maintain exorbitant lifestyles on the backs of an extremely poor and undeveloped country, so their solution, rather than developing the economy to increase long-term tax revenue, was instead to basically hold clearance sales of the entire country. The Reuter concession is an almost unbelievable example of it. They tried to sell essentially the entire country's economy to the guy who founded Reuters news: trains, trams, roads, forests, mines, canals, irrigation systems, telegraph systems, mills, factories, workshops, and even the national bank - in exchange for cash in the shah's private bank account. It was so bad that even the colonial powers of Britain and Russia said it was too much and wouldn't allow it.

The people were so pissed off about this sort of behavior that after they sold off tobacco rights to the British in 1890, there was a widespread boycott of tobacco, possibly the most successful boycott in history encompassing people across all of society, with a major religious leader even issuing a fatwa against violating the boycott, and going all the way up to women in the shah's own harem participating in it, forcing the shah to break the deal. The success of this mass action, combined with dissatisfaction with the shah selling out the country, led people to rise up in protest and start demanding things like a constitution and parliament. They even got the shah to sign off on it, and they brought in an American named Morgan Shuster to reform the tax code and root out loopholes and corruption. But when the shah died, his successor had other ideas, and he called in Britain and Russia, who were making quite a bit of money off loopholes and corruption, to come in and shell parliament.

Fast forward to WWI. Once again backed by Britain and Russia, the Ottomans invade Iran, and conduct the Armenian genocide. At the same time, there's a major famine, and the Spanish Flu is running rampant. When the dust settles, the old dynasty is gone, and a new shah comes to power, backed by Britain: Reza Shah Pahlavi. He attempted to push back against the oil deal but the British stonewalled him, and he lasted until WWII, when the Allies invaded to establish a strategic supply line between Britain and the USSR and to ensure Iranian oil didn't end up in the Nazi's hands instead of theirs. Afterwards, Britain forced him to abdicate and put his son in power, Mohammad Reza Shah Pahlavi.

The new shah also attempted to renegotiate the oil deal, and again was stonewalled. British advisors hand-picked prime ministers for the shah to appoint, in the mostly powerless parliament, but even then they stonewalled their own prime ministers when they attempted to negotiate. Public sentiment grew increasingly heated, and the advisors sent word back that the situation was reaching a boiling point, but still the British wouldn't move an inch. Finally, the shah became more afraid of a revolution than he was of the British, and he appointed Mohammad Mossadegh as prime minister, with a huge wave of popular support calling for nationalization of the industry.

Still, Mossadegh attempted to negotiate, and the US under Truman attempted to intervene as a mediator to prevent a crisis. But the Iranians had been struggling against British colonialism for a very long time, and the British still wouldn't budge. The oil industry was nationalized, and the British responded with a naval blockade that shut down Iran's economy and caused rampant poverty. Churchill approached Truman asking him to use the newly formed CIA to oust Mossadegh, and Truman told him to go to hell, that Mossadegh was a legitimate democratic leader and that the CIA was supposed to be about stopping communism not enforcing colonialism, and that the British brought this on themselves. There was no real animosity between Iran and America on either side at this point, the Iranians saw America as well-intentioned but naive, while Americans saw Iran as sympathetic but stubborn.

But Truman was replaced by Eisenhower, and Eisenhower was not acquainted with the background of the conflict, and Churchill and Alan Dulles were both pushing him to do it. Churchill changed tacts from talking about protecting Britain's property to warning that Mossadegh might end up aligning with the USSR, and also saying that he wouldn't support the formation of NATO or the Korean War unless Eisenhower gave him Iran. So he signed off on it.

They took over nearly every media outlet in the country, they bribed anyone who would take it, from politicians to vote counters to religious leaders, the hired protesters to march against the government and false flag "pro-government" protesters to go around wrecking things. Just before the coup, the American ambassador called up Mossadegh and told him a false story, that he was worried he would have to shut down the embassy because Mossadegh's supporters kept calling to give death threats, even to children at the embassy. Moved by the story, and eager to improve relations with the US, Mossadegh sent out a radio announcement telling all of his supporters to stay home and stop causing trouble. That's when the CIA made its move and deposed him, and no one was there to come to his aid.

They covered up their involvement for decades, until it was far enough back in history that people wouldn't care that they could finally admit it. That was the first democratic government they overthrew, and once the precedent had been set, they repeated around the world, Iran on behalf of BP, Guatemala on behalf of Chiquita, all across South America in Operation Condor, and so many more.