It's possible that he was very unpopular. I'm pretty sure it's confirmed from CIA papers since released that the mobs of protesters were organized, paid for and in some cases paid by the CIA, so there's at least an argument that his unpopularity was purchased.

The protests against Mossadegh included tens of thousands of Iranians. Is it possible to create that kind of unpopularity out of thin air using money?[1]

"A startled CIA reported to the White House that "an unexpectedly strong surge of popular and military reaction to Prime Minister Mossadeq's government has resulted, according to the latest dispatches from Tehran, in the virtual occupation of the city by forces professing their loyalty to the Shah and to his appointed Prime Minister Zahedi."

I'm not saying US didn't get it's hands dirty, but to assume that Iran would be a peaceful, democratic country otherwise is being naive.

[1]http://www.cfr.org/iran/myth-american-coup/p30900

I don't think they paid all of them, no, but I think it's entirely possible to fund a group of organizers who could motivate a significant-sized protest.

I make no assumptions about how peaceful and democratic any country in the Middle East would be, with or without US involvement.

So instead of electing someone else, they would install a shah?

The Shah already existed. He just used his power to dismiss Mossadegh (which was constitutional).