As for "no emergency and no crisis", Britain had organized an international embargo on Iranian oil, frozen Iranian assets, threatened military action, and was actively working to destabilize the government. Iran's economy was being strangled. Dismissing that as "no crisis" is basically ignoring basic historical facts.
On the military appointments, yes, Mosaddegh sought control over the military, breaking from tradition. But given that the Shah and military officers were actively conspiring with foreign intelligence to overthrow him (which we now know from declassified documents) his concerns about military loyalty weren't paranoia. They were correct.
The Hitler comparison is doing a lot of heavy lifting here. Hitler dismantled democracy and ruled for 12 years. Mosaddegh was removed by a foreign backed coup and spent his remaining years under house arrest. One of these is not like the other.
And in parliamentary systems, prime ministers who resign can return to power without new elections if they command enough support. It's happened in the UK, Israel, Australia and elsewhere. There's no constitutional rule that resignation is permanent and irreversible.
The question is whether Mosaddegh was constitutionally appointed and had parliamentary backing. In July 1952, the answer to both was yes.
Except that there was no legitimate emergency and no crisis, in fact there was just the opposite: Mossadegh had just instigated an insurrection against the then-legitimate government and thereby forced the Shah to not only put him back in power but also to let him appoint a defense minister and a chief of staff - a clear violation of the prevailing norms at the time which delegated this appointment to the Shah. The request for an explicit grant of power of "dictatorial decree" then came immediately after that. This was a clear established pattern of trying to weaken Iran's existing institutional norms and center power on himself, not unlike the whole Germany 1933 playbook. That's very much not how democratic systems work.