As for US support for the overthrow of Mossadeqh, it, like most of America's foreign interventions after WW II, was driven by fear of Soviet communist expansionism. Moreover, while the coup against Mossadeqh had covert US and British assistance, it was spurred and carried out by Iranian nationalists who -- like US and British country experts -- regarded Mossadeqh as an erratic kook and a national menace in light of the potent Communist movement poised to assume power when the Mossadeqh regime collapsed due to its gross misgovernment of Iran.
Thus, absent the coup against Mossadeqh and firm Western support for the Shah, Iran was likely to become a Soviet satellite. As it was, for the next two decades under the Shah, Iran prospered and Westernized. The Shah's eventual overthrow fits a familiar pattern of a restive populace overthrowing a corrupt and out of touch monarch or dictator.
I still say, regardless, it never would have become an “Islamic Republic”.