I dont want to Insult anyone, and i actually Think monarchy is a good thing for representative issues for a Country. Just like the Bundespräsident in Germany, just with some more Flair. But arent the Royals today more like a family-buisness with some Kind of "we make you more shiny" Deal with the british State? Thats at least what it seems like sometimes for outsiders.
The British Monarchy (apart from the period involving Cromwell) has over a thousand years of unbroken tradition. It is the centre of politics in the United Kingdom and a direct heritage to our foundations as a people, country and later an Empire.
Those who have studied UK politics know that the monarchy exercises huge political and symbolic power. The military swear to her their allegiance and upon my entry into the British Army I swore my allegiances to the Queen and her heirs not the government.
The Queen also maintains the UK's relationship with the Commonwealth as she acts as its head. Not to mention she is a rallying point to British, Australians, Canadians and New Zealanders, and as I mentioned with Trump, millions of people with British ancestry have huge amount of respect for her.
The Queen is also Head of the Church of England, and head of the British state and defender of the faith. She has ultimate authority but has continued the 20th century practise of following the government that can command the majority in the House of Commons. Ultimately for a bill to become law the Queen must pass her consent and sign it. There theoretically the Queen could refuse any law she didn't like and even refuse to sign a bill that called for a republic.
The Queen could easily refuse to sign a declaration of war and has issued resistance in the past.
Also, as British custom, every week the Prime Minister must report to the Queen in person, and inform the Queen how he/she is running the country and there reasons for it. In this private audience the Queen has the opportunity to be 'consulted, warn and advice' her minister.
Many worry that Prince Charles will not follow this convention and that may cause a constitutional crisis.