Liberal countries tend to befriend other liberal countries. So yes, even from a purely strategic perspective US foreign policy has a desire to liberalize other countries (in the classical sense of the term).
Moreover, multiple US administrations pressured the Shah on human rights and, more generally, the US has a definite (though certainly not absolute and varying by administration) preference for working with regimes with a decent human rights record.
I would also quibble with the sole attribution of his fall to Ajax (he was already deeply unpopular when it began, albeit arguably unfairly) and note that his extensive use of emergency powers at the time make the "democratically elected" bit somewhat misleading, even if nominally true.
Mossadegh was not elected by the people of Iran to be Prime Minister.
Mossadegh was not democratically elected as head of the government. He was put into that position by dictate of the elites that ruled over Iran in fact, including with the backing of the Shah. He was democratically elected to the Majles, which is not the same thing as the people electing him as Prime Minister. The Majles largely consisted of elites that were wealthy property owners. So the de facto ruling feudal lords of Iran - including the Shah - installed Mossadegh as leader of the country, the exact opposite of democracy.