On the recommendation of the PM, not on his own initiative - following the precedent of William IV in 1832 - in both cases essentially enforcing the will of the elected House of Commons against the unelected House of Lords
I agree with your cogent analysis of the power of the Crown, mainly that there isn’t much of it, except as you say in the sense of a collective conscience and moral suasion. That power, such as it is, exists only because the monarch is seen (or should be seen) as impartial and non-political. It’s useful to a society to have a presumably honourable and impartial figure as Head of State rather than, say, some political hack. In my view this is one of the strengths of a constitutional monarchy. It was well illustrated when Trump visited the UK. Britain was represented by their Head of State. the venerable and well-respected Queen Elizabeth. The US had the extreme misfortune to be represented by a clueless boor, because this bum was their Head of State.
But just a few comments on things that are Simply Not Done. The monarch or their representative in Commonwealth countries, the Governor General, actually does have – in theory – considerable reserve powers. They just have to be extremely cautious about using them lest they be seen as impeding the democratic process. For instance, no bill passed by Parliament can become law without royal assent. In a democratic constitutional monarchy, royal assent is assumed to be automatic, but in theory at least it can be withheld if the monarch or GG believes it to be seriously detrimental to the national interest. Similarly, the opening, closing, or proroguing of parliamentary sessions is at the discretion of the monarch. I’m not aware of any instances in Canada in which the GG refused such a request from the Prime Minister, but there were several instances in which it was a close and tough decision that could have gone either way.
And speaking of reserve powers, not all of which are enumerated:
Typically these powers are: to grant pardon; to dismiss a prime minister; to refuse to dissolve parliament; and to refuse or delay royal assent to legislation (to withhold royal assent amounts to a veto of a bill, while to reserve royal assent, in effect, amounts to a decision to neither grant nor refuse assent, but to delay taking a decision for an undetermined period). There are usually strict constitutional conventions concerning when these powers may be used, and these conventions are enforced by public pressure. Using these powers in contravention of tradition would generally provoke a constitutional crisis.
Reserve power - Wikipedia
One interesting final note pertaining to the matter of the Crown speaking to the public. Some years ago a Quebec radio DJ made a prank call to Queen Elizabeth pretending to be Prime Minister Chretien, and succeeded in getting through to the Queen. He made several attempts to put her in a difficult spot, and the one that’s pertinent here is that he asked her, in a tense period just before Quebec ran a referendum on seceding from Canada, whether she would make a statement in support of the government’s position to vote “no” on the question.
This really did put the Queen in a bind, as she was torn between deference to the Prime Minister and the tradition of staying above such political matters. Genuinely uncertain of what to do, she briefly consulted with an aide, and told the PM to send her a draft of the statements he wanted her to make. She appeared to be willing, albeit reluctant, to do it, and probably only even considered it because it was in support of national unity.
As an aside, the Prime Minister called the Queen later on to apologize for the hoax, and she was good-natured about it. Her dismay at this sort of request was evident, though, when she laughingly told the PM, “I thought you were drunk!”. 
IIRC, didn’t Her Majesty also respond to the prankster in perfectly fluent French?
Yes, that was another attempt to put her in a bind, but the prankster apparently didn’t know that she had been well-trained in French. I remember listening to the audio of that call, and to my ear, her French was indeed perfectly fluent.
Yes, that joke kind of backfired on the caller. And although I don’t know for sure, I wouldn’t be surprised if Charles isn’t also fluent in French. If Elizabeth is any indication, it is likely part of any Royal’s education.
True, but as for me, I’m only half American—I’m also a Canadian who lives in Canada, and having lived a considerably number of years in the US and in Canada as an adult (and some in the UK), I think that on the whole the Westminster-style democracy, monarchy included, is a better system. The only change I’d seriously propose would be to make abdication a realistic option for those who don’t want to do it, or those who would like to retire. I understand why that’s not currently on the table, though.
Although perhaps not German. Some political third rails are still warm from the last conflagration.
Charles speaks German fairly fluently though I’m told his accent is not great. IIRC, has given public speeches in German, in Germany.
He has lots of German relatives on his father’s side, remember.
A SD Staff Report (remember those) on the episode, by our esteemed colleague, @Elendil_s_Heir :
I thought his father was Greek, not German.
Exactly my point. The Windsors have been trying to downplay their inherently German roots since about 1933. To varying success. And with varying degrees of give-a-shit-about-that in return from the public.
According to Google, Charles speaks French, German, and Welsh fluently. I’d imagine that he probably has a rudimentary amount of training in the languages of some of the other Commonwealth realms, so that if he were called on to make a state visit to, say, Tuvalu, he could at least say a few words in the local tongue without sounding like an idiot.
It’s…complex.
Yes, Philip’s father, Prince Andrew, was the Prince of Greece (and Denmark), but he wasn’t Greek by background. Andrew’s father, George I, was a Danish prince, and was elected to become the King of Greece by the Greek National Assembly, after they had deposed their previous king, Otto (who wasn’t Greek, either; he was Bavarian).
Prince Philip’s Danish bloodline, though his father’s and grandfather’s line, was intertwined with various German noble houses.
In the same way, Charles’ mother was German.
Right. They’ve changed their name, but they’re still basically Hanovers. And Charles is descended from Victoria on both sides.
There were a lot of German royal families, so in the time when royals marry mostly married other royals, every royal family was constantly acquiring German in-laws, which turned out to be inconvenient in the 1910s and 1940s.
Quite.
“The Version I Heard” was profoundly wrong in a virtually every regard.
It hasn’t even been determined whether the Head of State of the Australian republic will be called “President” or “Governor General”.
And as noted, despite “The Australian Republic” probably still holding majority support as casually polled, the Australian Republican Movement is currently moribund and couldn’t win a plebiscite, or a chook raffle.
When in Papua New Guinea in 2012, the then Prince of Wales used Tok Pisin to describe himself as “nambawan pikinini bilong Misis Kwin”.
Best line-The British monarch does have what historian Robert Lacey calls a “dry and slightly mordant sense of humor,” and has in recent years developed a stock line for whenever the cellphone of a visitor goes off: “You’d better answer that. It might be someone important.”