Not an expert perse, but that is indeed more or less how it works since 1848 in the Netherlands. In Belgium it exists since 1831 though Belgian members-of-parlament already tried to pass it in 1815, still with the house of Orange. It supposedly is based on the works of Benjamin Constant.
The Prime Minister is responsible for wat the King does and needs to explain this in parlament, thus also curtailing the power of the King. The idea was at the time that it would work as a sword and a shield. A sword in the sense that parlament can democratically check the King. And a shield in the sense that the PM will protect the King. It works the same way for the civil service at the ministeries and the responsible ministers.
Whenever there is something with a royal the PM needs to come to parlament to explain. In the 2000s we had among others the COVID trip of the King to Greece, The King and hunting, Laurentien and her foundation, role of Prince Constantijn in Techlab, sale of art work by Juliana's daughters, Bernhard and his SS membership, Bernhard and his use of the secret services, Bernhard and an alleged attept at a coup d'etat in Indonesia, many financial incidents (Beatrix' sail boat, income Amalia, salary increase, restoration palaces, letter box firms for Irene, Margriet and Christina and families etc) the house in Mozambique, the house in Greece as well as more seriously Mabelgate, father Zorreguieta and Margaritagate -the later was an absolute cricus.
I am always amazed how other monarchies seem a-political to a large extend and function in a vaccuum. A scandal like Andrew, Mette-Marit, Martha-Louise or even the Sussexes and their accusations would all have been fodder for parlament here. Both Andrew and MM with the potential of a real political crisis. The only other monarchy where I see a PM regularly answering questions about the monarchy in parlament is Belgium.
It has positives and negatives. The positives are that it gives the government, parlament and thus the people a say in how the head of state functions. It protects the King [all he does is the responsibility of the PM]. On the minus side it that things get politisized quickly, sometimes for political gain, more to hurt the PM than out of concern of the issue itself.
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As for the distinction of the royal house and the royal family. The law limiting the royal house dates from 2002. The reason was that the family was expanding and it was impossible for the PM to be responsible for a dozen or more people. Many of whom wanted to pursue their own lives and carreers. Informally it was said that the Windsor soap operas of the 90-ties made Queen Beatrix realise that a clear distinction needed to be made. It was clear to her that the supporting cast could damage the institution while little they did would significantly improve the image of the monarchy. To underline the distinction she gave her own grandchildren from her younger sons different, lower titles too.