"I think that there is something admirable about the governments of the U.K. <...>; France, Germany and Italy could plausibly benefit from their example."

I don't disagree with your assessment but ultimately the writing is on the wall for monarchy (as, on average, the powers of monarchies have been atrophying steadily over the last few centuries and there's no sign of any letup). Much of monarchy's current appeal in countries that still possess monarchies and the concomitant [apparent] stability of their political systems comes from both nostalgia and a reticence to change (given historical circumstances, such views are hardly surprising, especially so in countries with a long and illustrious historical (monarchical) lineage).

One wonders how, say, the U.K. would survive in a cultural sense if its monarchy were to be suddenly abolished. After all, its cultural heritage is extremely important to a huge number of its citizens, especially so given its loss of empire (from which the country has been smarting badly ever since WWI and especially so since WWII when in the wake of the War the country suffered significant loss of power and prestige on the international stage).

Nothing could have made this more obvious than The Suez Crisis of 1956 when on the world stage the U.K. looked like a petulant brat sans any diplomatic experience (one even cringes when thinking about what happened). Whilst the U.K. has largely recovered from this hugely embarrassing incident and that it's largely forgotten by many of its citizens (likely because of its embarrassment), it nevertheless only ever achieved a partial recovery in status.

(The evidence for the decline in power and influence of monarchies is reasonably solid. One cannot forget that monarchical power was once essentially absolute but these days (fortunately) it's only a shadows of its former self. It's impossible to put aside or forget the maniacal malevolence of Henry VIII or say the utter hatred George IV's subjects had for him. The fact is that whilst the monarchy represents stability in some countries it doesn't necessarily follow that it will always stay that way (and as we've seen with pre-WWI European monarchies, Hapsburg, etc., situations can change dramatically in just the spur of a moment).

I'm not condoning such rapid political change, as it inevitably comes with considerable dangers as witnessed by, say, the French Revolution. Arguably, the Reign of Terror - la Terreur - that followed the French Revolution and the beginning of the First Republic has few equals in modern history with respect to its sheer brutality and its effects have been very long lasting indeed (as my French relatives keep telling me and demonstrating the fact with any number of present-day instances). Thus, in this regard, your comment about France, etc. plausibly benefiting from the example is rather apt; nevertheless, I don't see that detracting from my argument (as it's a separate issue).

That said, the English were hardly any better, it's just that they started earlier. The Battle of Towton in 1461 during The War of the Roses, is almost unbeatable for sheer violence and utter brutality (although it's not alone with such a reputation): https://en.wikipedia.org/wiki/Battle_of_Towton.

I mention this and hold my views about monarchy not from any gloating republican perspective—in fact far from it—but rather from evidence that supports the position that monarchies are in decline. Of course, many could or would contend that I could be just saying this and that in fact I'm a diehard republican at heart—and I acknowledge that it would be difficult for me to disprove the fact. In my defense, I can only state that I'm from a Commonwealth country (Australia) and that I've U.K. parentage. Moreover, the fact that I'm still in possession of souvenirs of QE II's coronation that were given to all of us children when we were in infants school signifies its importance during my early formative years (yes, the propaganda began early and it was very effective; back then we kids had no doubt that the U.K. was the greatest country on the planet—even though it wasn't our native homeland)!

Here's a reference to one of those souvenirs: https://picclick.ca/1953-Queen-Elizabeth-II-Coronation-Memen..., the other one I've not yet found any reference to but it was about the Crown Jewels and described the various items that comprised the collection together with an explanation about each (the meaning of the different crowns, sceptres, maces, etc.).

Incidentally, given the once ubiquity of these brochures in that they spanned multiple Commonwealth countries, I'm somewhat surprised how hard it was to find any reference to them on the internet.