Today all political interactions are purely ceremonial. The monarchy is still part of the power elite, and of ceremonial importance in domestic and international relations, but they have no direct power over political decisions or elections.
[1] https://en.wikipedia.org/wiki/Christian_X_of_Denmark#Easter_...
The 1975 Australian constitutional crisis, where the Governor General (granted power by the Queen) dismissed the Prime Minister.
In 2019, when the British PM suspended Parliament - a provision that would not exist without a Monarchy, as it was technically done by the Queen on the PM s advice. This was later ruled illegal but there was no way to right the wrong.
Recently, Queen's Consent was exposed as a secret process where laws are changed in secret before coming to Parliament to gain the Queen's approval. One example being that the Queen can hire without complying with laws against race discrimination. She also has exceptions from environmental and COVID legislation.
Checking Wikipedia, it looks like Queens Consent was withheld in 1990 preventing debate of reforming the House of Lords. Probably the world's least democratic legislative chamber of any democracy.
Queen's Consent is a better argument, although worth noting that it's more like "I don't have to follow your laws" rather than "I get to change your laws".