The CIA, in its own words, seems to feel that it was responsible for the "overthrow" of Mosaddegh, in a "coup" (their words).
At the time of the coup, Mosaddegh was more popular than before, enjoying majority support, and had just been granted emergency powers by parliament for six months, which then renewed them for twelve. I respectfully dispute your suggestion that, absent CIA interference, it's doubtful that he would have maintained rule.