For context:

Yeah, we can totally trust a bunch of hateful gusano sellouts to bring glorious liberal parliamentary democracy to Iran. Them pissing all over the Constitutional Revolution’s legacy and thinking some glorified warlord and his spawn have a God-given right to override democratic decision-making, via the only representative avenue the filthy common folk had to express themselves, is entirely incidental (trust them also to bring up the fact that Mossadegh eventually dissolved the Majlis without a hint of irony).
Not to say Mossadegh did everything right (he trusted the US not to screw him over, after all) but to these parasites trying to smear him, his unforgivable crime was attempting to run an actual liberal democratic government and asserting Iran’s sovereignty over its own resources, like a certain other country the empire is in the process of besieging. Also Mossadegh becoming more well-known figure throws a wrench into the image of the Pahlavi family being glorified as national modernizers planting the seeds of “European-style” democracy.
I’ve seen some lib Westoids parrot this and it makes complete sense. It absolves them of the responsibility for helping strangle a fledgling liberal democracy in its cradle and supporting the modern day ideological descendants of its butchers. It also helps them sidestep the fact that the legitimacy of the Iranian government is rooted in their perceived liberal nationalism and anti-imperialism just as much as it is in the Islamic faith. Makes it easy to swallow the slop that the Mullahs and their IRGC enforcers are insane fanatics who must be prevented from having WMDs (which definitely exist) or any real military capabilities if not immediately toppled at all costs.
Isn't the UK prime minister a similar situation? Appointed by the king?
This is the case in all European countries with royal heads of state, meaning Denmark, the Netherlands, Norway, Spain etc.
Formally/de jure yep
De facto they likely couldn't go against the results of a generally election without pushback from parliament and the public(regardless of the extent of their real hard power over the state they still have massive influence over most of it's organs, including those they're officially head of such as the military and Anglican church), so monarchists use this to excuse them from responsibility for any war crimes etc committed by the rest of the state, similar to the people who argue Hirohito had no responsibility for Japanese actions during WWII
A few times in the 19th and 20th centuries, the UK monarch has appointed a PM without a corresponding election or Parliamentary vote. But only in the 1800s did they specifically go against a vote.
As you say, they still exercise their power in very real, other ways. The UK monarch has been shown to secretly vet, veto and amend laws all the time, at least around a thousand over the past few decades. And those are just the times we've found out.
You'd think so, and I don't think anyone of them tried, but as a counterpoint: Macron (who is as close as you can get to a King of France without being one at this point) did pretty much that multiple times and nothing happened.
Yeah but that's because in France, the president is the head of the executive, not the prime minister. The PM is chosen by the president, but every executive functions reside with the president, who also can dissolve parliament and call a new election or sack the current PM to choose another from the majority party in the legislative assembly.
It would be interesting to see that come to a head, but with how reactionary the UK government already is, what could the monarchy possibly gain from opposing it? Now that I think about it though we would definitely see the Crown testing its powers if a leftist government ever got in.
We certainly saw the Crown test a lot of unofficial powers when Corbyn, an anti-monarchist, was a potential PM.
The David Weber fantasy of the woke Crown allied with the Commons against the greedy and venal Lords
Also not even all lords, just a few bad apples.
Ah, good Tsar and bad Boyars. Timeless classic.
I hadn't heard that term for it, so I did a little search. Fun how NATOpedia seems to think it's almost exclusively a defect of the Slavic brainpan.