Interesting, but not entirely accurate. The Queen herself, Mrs Elizabeth Mountbatten, has no power or authority except over htose she directly employs.

All the powers you have just described belong to the insititution of the Crown. The Crown is not the person of the monarch. We could remove the Mountbattens and restore the Stuarts (Franz, Duke of Bavaria being the current Stuart heir), search for an heir to Harold Godwinson and restore the English monarchy, or go even further back and hold an Althing to elect a monarch! None of these would change a damn thing, any more than the election of a US President changes the Constitution.

Because that is what the Crown is, our equivalent of the Constitution, the source of authority for the government. The House of Commons gets its mandate from the electorate, but its authority from the Crown. The House of Lords gets its membership via the peerages granted by the Crown. Civil servants are employed by the Crown. The judiciary and the Courts derive their authority from the Crown, as do the police. This is designed to keep them politically neutral and to give them a legal right to refuse to serve an openly corrupt or tyrannical government or monarch.