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.