I searched the article and its reference for any details on the mechanics of this "coup." Zilch.
WSJ has an article today, in fact, about it. Mossadegh was not "democratically elected." There was massive opposition to him inside Iran. The UK was opposed to him, of course.
As for the "rented" crowds: evidence please?
> IRAN 1953: MI6 PLOTS WITH ISLAMISTS TO OVERTHROW DEMOCRACY
> Declassified British files highlight a little known aspect of the joint MI6/CIA coup against Iran’s democratically elected government in August 1953 – UK covert action in support of leading radical Shia Islamists, the predecessors of Ayatollah Khomeini.
That's why in 1956, the UK and France had to humiliatingly climb down from the Suez operation. Eisenhower just said, nope, nope and that was it.
See also: presidents of the United States of America
There is no contradiction between democratically elected and domestic opposition. Quite the contrary.
The clergy were against him because their agenda didn't align with his progressive one.
Massive opposition does not mean not democratically elected.
Trump case in point.
Btw, "Shah" means king. Iran had deposed a monsrch, that was part of why the Brits did not like it.