After President von Hindenburg's death, there was no one left in Germany for Hitler to answer to, not even in theory. That meant there was no legal mechanism to remove him, he could only be removed by violence – and the plot to assassinate him failed. The world could easily have been a better place if Germany had remained a monarchy – even if Hitler still became Chancellor and started World War II and all the rest of his horrid crimes, it would have been far easier to remove and replace him with a new German leader. Germany's generals knew they had lost, and many of them wanted to focus on making sure they surrendered to the Western allies not the Soviets - but Hitler forced them to keep on fighting unwinnable battles instead. With Hitler imprisoned (or maybe even summarily executed to eliminate the risk that devout Nazis would try to free him), the generals could have done as they wanted. Possibly the division of Germany could have been avoided, and large swathes of Eastern Europe could have escaped communism. It also could have resulted in much better treatment of German civilians and POWs, for while the Western allies weren't perfect in that regard, they were significantly more humane than the Soviets were.
von Hindenburg being President or Wilhelm being Kaiser changes little assuming a death - in both cases there's a process to get someone to replace them (new elections for President, coronation of the heir apparent). Hitler's popularity and power allowed him to skip the whole process and just assume the position of Parliament too. I think that it would have been harder to do with the position of Kaiser (there were still plenty of monarchists even in Nazi Germany), but he still would have done so or at least done a Horthy, preventing the coronation and keeping himself as regent.
There were actually numerous plots to assassinate him; the July 20th plot is just the most famous (probably because it came the closest to success).
This guy came very close just a few months after WWII started: https://en.m.wikipedia.org/wiki/Georg_Elser
Japan also had a monarch, and it took 2 nuclear bombs to get him to surrender.
Though nothing is stopping the cult of personality from forming around the leader of the government (prime minister).
Not really. The cult gives its focus power regardless of the formal structure; that's the point. And in the case of England, the formal structure still supports the king! Forming a personality cult around the King of England is more likely to lead to the King of England reclaiming a lot of power people thought he had lost long ago than it is to defuse the cult when they learn the king isn't supposed to use his formal powers.
I assure you there's nothing even CLOSE to that insanity in European monarchies.
Edit: I haven't been able to turn up good numbers for the original cost of Versailles without spending more time than I have to spare at the moment, but on an ongoing basis the annual budget for Versailles is about €100 million, while the annual budget for Mount Rushmore is about $5 million.
Edit #2: Looks like the Brits spend around £50 million annually for the maintenance of royal palaces, and those are still in active use.
Which is great! There's enormous value in having a ceremonial national institution which isn't politicised and doesn't split opinion along the typical partisan lines. It's a part of our national identity, something that we all have in common, something that unites us - and in these polarised times we desperately need more things that unite us.
If we replace the monarch with an elected head of state, even a "powerless" and "ceremonial" one, there's no way it won't lead to more polarisation and division.
The other good argument I've heard for keeping the monarchy is that abolishing the monarchy means rewriting the British constitution from the ground up. You can be sure that whichever party is in power will take this opportunity to subtly tweak things to favour themselves. Do you trust them to do it fairly?
Better the devil you know.
It's possible (but admittedly difficult) to construct a mythology where you make a rock the national identity, and celebrate the rock on special occasions. It lasts for thousands of years, you don't run into problems of succession, and it never betrays anyone, much less express political opinions.
So, why don't you make a rock your King? It seems to fit all your arguments about why a human king is good, and the rock does the job better.
PS: apparently in the US the equivalent is a piece of paper called the constitution. The president, despite what monarchists think, is generally disposable and often just for lols.