The PM bows to the Queen, but that doesn't mean they have to listen to the people more than they do in France, no?
Doesn't the French Prime Minister answer to the President? How is that worse than having a monarch? Are they often from the same party, thus rendering this answering to the president less powerful? (I know the current PM and President are, not sure if that's the common case.)
My impression is that just by being less involved in politics, and generally (not 100%) staying away of corruption and other sorts of scandals (unlike others, looking at you Juan Carlos I) for a few decades, the figure of the Queen can be less jarring or seems more trustworthy than a President usually would.
To be honest, I live in a monarchy, and if I could choose we'd transition to a republic... but I've never felt like it would make a huge difference in the quality of our government or our electoral politics, so I just don't really care.
In theory, no, French PM answers only to Parliament. Only Parliament can dismiss them, not the President.
In practice, and in normal times, this isn't true at all. When the President tells the PM that their time is up, they immediately resign. (One tried to resist in the 70s and was immediately voted out by Parliament.) This makes the French PM effectively powerless. They simply implement the will of the President. The equivalent to the British PM is the French President, not the French PM.
Now there are non normal times where Parliament and the President are on opposite sides. When that happens (1986-1988; 1993-1995; 1997-2002), the PM is effectively in charge of most things, but even in those cases the President still has more powers than an typical constitutional monarch.
But my point wasn't about power but about humility. I think it's good and desirable that the ruler has to bow to someone else, and that that person, in turn, has no power whatsoever.
The queen can still secretly prevent legislation from ever being heard in parliament, so...
The point is that a president is a) elected and b) works openly.