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?
Her power is absolute and temporal, she isn't a mere figurehead. She does wield this power from time to time, she dissolved the nation of Australia not too long ago on a whim.
I think that grossly misunderstands what happened in Australia with the Whitlam government dismissal.
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.
Almost 50 years ago. Caused a constitutional crisis. Would not think it’s repeatable elsewhere in the commonwealth. Reasonable chance her successor will be head of state of fewer realms. Constitutional monarchs use their power at risk to their institution. Sometimes it’s quite useful (Spain), but it would quickly fade if used repeatedly.
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.
Australia has enough tanks. In case the Queen want to dissolve the government they can just tweet "Nah" and continue as usual. What is the United Kingdom going to do? Invade? Ask Canada and New Zeeland to make a joint force?
(Officially breaking will cause some problems, so it depends a lot on the internal situation and how much popular support the government has.)
Rereading the wiki article, it seems Australia resembles England before the House of Lords was broken in the 1905-1909 time period - the upper house can influence money bills. It would seem that this was a case of the Governor General involving powers of the crown that it has but should not use. The gg was Australian.
Perhaps this is (another) example of the difficulties of running a government composed of more than one centre of power.
To understand just one of the Monarch's many powers as the representative of the Crown, you should read the essay entitled The Royal Power Of Dissolution. There is no "normal" limit, the Monarch, acting on behalf of the Crown, is Sovereign.
I think a set of individuals having legal privilege by virtue of bloodline, as well as being state sponsored by tax revenue (regardless of absolute cost), is pretty anti-democratic. And yes, elected is literally democratic?