The Shah had re-instated the same Parliament that his predecessor, Mossadeq had dissolved, while Mossadeq was well on his way to becoming a communist-leaning dictator.
The form of Parliamentary Monarchy re-established by the return of the Shah, is much like the British model was, in times past.
I say was, since now, the British Monarchy seems as much a figurehead, as anything else...
Today, the world is quiet a different place. There's nothing at special about being a "Republic" anymore, since the vast majority of nations have set up their states as Republics. There are 135 Republics in the world today. Whether a Republic is "good" or not depends on what type of government they have. In Iran's case, they established an oppressive theocratic Islamic Republic, so the case can be made they were better off under a figurehead monarchy that was friendly to western nations and respected human rights. I would also say France was a far friendly ally to Americans when they were a Kingdom than any time during their status as a "Republic"
Likewise, having a "King" does not bring forth the dreaded fear it did 300-400 years ago, since most of the remaining monarchies in the world have replaced their form of government so the "reigning monarch" is only a figurehead. As you noted, most nations like Iran just followed the British model. In today's world, the number of remaining monarchies where the King has strong control of the nation's affairs can be counted on one hand. I believe the current number of absolute monarchies in the world consists of three nations.
The United States is not an oppressive dictatorship precisely because we have a much stronger democracy than Iran does. Therefore, the freepers who claim democracy is bad and leads to "mob rule" have it backwards. These "conservatives" sound like communist overlords of China the way they scorn about how terrible it is to let the public decide things. The United States is a representative democratic republic, and that's why we enjoy freedoms that the theocratic Islamic republic of Iran doesn't. Their amount of democracy is extremely limited and subject to the Ayatollahs whims.
The freepers who claim that a "Republic" and a "Democracy" are separate, competing forms of government are also talking the B.S. The United States clearly has both. Saying you can't have democracy and keep your Republic is like saying you can't have capitalism and keep your Republic. "Democracy" is a type of government. "Republic" is a type of nation-state. It's perfectly possible for both to co-exist within an organization country.
It's true that DIRECT democracy leads to "tribal rule", but since no modern civilized nation on earth has established "direct democracy" as their form of government, the idea that the U.S. in "danger" of becoming one is ridiculous.