Fast-forward to the end, if you're impatient, to see quotes which should make you see the Queen in a different light.
Britain is an old country - one of the few real old countries in the world. (France has had a monarchy and 5 different republics in the last, mere, 200 years; Italy - monarchy, fascism, republic; Spain - monarchy, running alongside fascism; Germany - didn't even exist as one country 200 years ago...separate states, subjugated by Napoleon, then Empire, democracy, dictatorship...anyway you get the idea: other countries have changed a lot and Britain hasn't.) If you did some research into how this country which hasn't changed much since 1066 AD (our last foreign invasion) works, you would be astounded at how critical the monarchy is. By that, I mean it's a cornerstone: just take it out and the whole lot comes crashing down. You could turn Britain into a republic, but you would have to change it gradually and fundamentally. Look at Russia when they tried to jump straight from communism to capitalism.
The Lord Chancellor, a role older than Prime Minister (PM), is the head of the judiciary and one of the most important leaders of the House of Lords (an unelected political chamber filled with wise old dudes and dudettes - including the top judges). The PM recently, selfishly, felt it would be better to scrap the position of Lord Chancellor. So he did, practically overnight. It was a disaster - for the main reason I outlive above: you do not take away a cornerstone suddenly. The position was resurrected. Gradually, now, it is being phased out.
One other point. Historians now consider monarchies as stable or more stable than republics. Why? Because a monarch's time-horizon tends to be far longer than an elected politician's (who is only looking 4-10 years ahead, when his re-election is due!). Of course, you can have a bad monarch on the throne, but you can also have a bad President (Nixon anyone?). The Harvard historian, Niall Ferguson, admittedly British, is of the opinion that the US is the only republic in the world where the 2000 presidential stalemate could have occurred without a single shot being fired.
A monarch with some powers, who has a 100-year time horizon, and a Prime Minister who has, say, 10-yearly ones (two elections' worth) is what we have in the United Kingdom of Great Britain and Northern Ireland and it works pretty well.
You say: "British citizens have to pay millions of tax dollars whether they like it or not to keep them around".
The entire British monarchy costs less than the US President's Air Force One, alone. Also, the monarchy is a huge tourist attraction and earns the UK a lot more money than they cost. How many tourist bucks does the US President tend to earn his country? A negative amount, probably; but, to be fair, he saves them a lot through cheaper oil.
They're not just figureheads. Take the Queen. Elizabeth II is Queen of sixteen sovereign states, holding each crown and title equally. However, she is more directly involved with the United Kingdom, where the Royal Family resides, and the Monarchy is historically indigenous. She is Queen of Canada, Australia, New Zealand, Jamaica, Barbados, the Bahamas, Grenada, Papua New Guinea, the Solomon Islands, Tuvalu, Saint Lucia, Saint Vincent and the Grenadines, Antigua and Barbuda, Belize, Bermuda, and Saint Kitts and Nevis, where she is represented by Governors-General. The sixteen countries of which she is Queen are known as Commonwealth Realms, and their combined population is 128 million.
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Some quotes of hers:
* "Grief is the price we pay for love."
In a letter read at the memorial for British victims of the 9/11 attacks held in New York.
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* "I do hope you can see me today."
Opening a speech to the U.S Congress on May 16, 1991, the day after giving a speech where the podium was too high for her to see over. This comment generated gales of laughter and a standing ovation.
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While being shown around an artificial insemination unit of the Milk Marketing Board
"What's that?"
"It's a cow's vagina, ma'am"
"Ask a silly question!"
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In reply to a shopkeeper who said, "You look awfully like the Queen"
* "How very assuring!"
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And finally:
Said to a young lady (who was invited to have tea with her) when the young lady's mobile phone rang during their talk:
"You'd better answer that. It might be someone important."