We live in very uncertain times today, and as Zonk said above, in too many countries there is an increasingly polarised political situation, even in the best democracies. In such times, the importance of tradition and institutions above politics is paramount, which is where the value of monarchies in countries that retain them are.
No few of us, and certainly many of our parents and grandparents (who in turn had it passed down from their parents and grandparents), can remember anywhere between 1914 and 1989 when, let's face it, Europe was more or less at war with itself with two World Wars and the Cold War. This came after the so-called "Long 19th Century" between 1789 and 1914, the period between the French Revolution and World War I, that saw profound changes in the social and political makeup of Europe. It's in this context that we come to realise that too radical a change can be a bad thing for everybody.
And we can also remember the worst abusers of human rights, apart from absolute monarchies in the Middle East, were the dictatorships of Latin America and Eastern Europe, and those weren't monarchies. I guess from seeing how those regions in particular are still coming to terms with the mess of the 20th century, we can appreciate what we have here and now.
You can't say that Germany and Portugal were exactly better off after their monarchies were overthrown. It took many years for Portugal to evolve into a stable democratic republic, and that came after 50 years of dictatorship. Similarly in Germany and Austria, you have some historians arguing that if the monarchies had survived, they might have prevented the rise of Hitler and National Socialism, though I'm not sure if everyone would agree with that one.
Maybe it's in this context, that people are reluctant to change what they have now.
