Australia had an example of this where the Governor General (the Queen's representative in the Australian government) sacked the government and formed a caretaker governmen tin the 1970s.
The American system of relying on centuries of tenuous interpretation of a fairly short document just isn't as much of an improvement as you think.
The Australian constitution allows for the Governor General to have "reserve powers" without specifying what they are.
Ultimately all these systems rely on trust.
It's wild to think the Queen began her reign with having weekly chats with Winston Churchill all the way up to appointing Liz Truss just this week. Her father fought in World War I. She lived through World War II. It's wild to think about.
It's also wild to consider the Queen never had an exepctation of ruling. An abdication caused that to happen. The happiest and freest time of her life may well have been living on Malta prior to that, living a fairly normal life with her husband and young family.
Institutions exist to protect the people, not the institutions themselves. Never forget that.
Having an unelected, unaccountable individual who leeches off the tax system: this is anachronistic but fair, it's about balance of powers, its an important part of our cultural heritage, it doesn't even cost that much why do you care.
Having strong founding principles and rights that are cautiously amended: this is tenuous, this goes too far, free speech too extremist, why bad man own gun.