How is the UK monarchy anti-democratic in any manner? She has only symbolic governmental power. Alternatively, do you feel the monarch should be elected for some reason and if they were, what problem would that solve?
Analogous to any non-executive head of state I would think. However, the time of a non-resident monarch has passed - they should be of and reside in the country they're head of state of.
Perhaps this is (another) example of the difficulties of running a government composed of more than one centre of power.
(Officially breaking will cause some problems, so it depends a lot on the internal situation and how much popular support the government has.)
The queen has been effectively stripped of all personal discretion when it comes to the use of her reserve powers.
She has no real choice when it comes to who is appointed prime minister or which bills are made into law.
The voters and elected officials make those decisions, and her only real job is give a royal stamp of approval.
The executive power in Australia is held by the Governor General who "represents" the Queen, but is limited by the constitution in what powers can be exercised. The Monarch of Australia essentially has no power in Australia. The Governor General can largely only do what he/she is told by parliament or resign. The "crisis" part here was that the Governor General found a grey area, the current Prime Minister could not form a majority in the lower house and so the Governor General appointed the opposition leader as the Prime Minister who immediately asked the Governor General to dissolve parliament. It was dubious that the Governor General could actually do this, and it has since been codified in law that he/she cannot.
The Governor General did discuss the matter with the Queen, but largely she had no part in this decision. In fact the released correspondence between the Queen and Governor General shows the Queen declined to make a decision.
There are many people in the world who currently have some form of legal privilege by virtue of bloodline. For example, many indigenous people.
I don't want to get into a tit-for-tat about this. For me, at worst The Queen seems benign at best a proud symbol of a great nation.