Do "Western democracy" and "European-style monarchy" refer to single things over the course of whatever length of history you are considering? The feudal systems of the Middle Ages, Britain in the early 19th century, and a large part of Europe under Napoleon are very different monarchies, as are Britain in the later 19th century and present day USA different "Western democracies". Does Spain under King Juan Carlos fall into both classes?
Even if one chooses single paradigmatic instances of "Western democracy" and "European-style monarchy", the question begs the question of whether there is such a turning point. South Africa is a Western(-style) democracy but doesn't seem to be doing very well at it, and there are people today who would prefer a system of "everything for the people, nothing by the people". Moldbug, for example. Or the NSA.
These are some of the questions that a highly rational actor would have to think about to make the original idea meaningful.
Some time before 1789.
One of the key steps in the lead-up to the French Revolution is when Louis summons the Estates-general. And he does this because his ministers say "look, if you want to fix the financial mess that the state is in, you need to run the country more like Britain, with a parliament that can make credible promises." Even many of the skeptics and opponents of the French revolution, such as Burke, agree that parliamentary government is the way to go -- they just did't think France could safely get there.
The conclusion I draw is that by 1789, a lot of smart serious people whose careers are at stake had gotten convinced that the absolute monarchy of the Bourbons was irreparably broken and needed radical change towards democracy.
There's an additional question, which I think matters more in practical politics, which is "when did democracy become a viable option in various countries?" The usual defense of monarchy isn't "it's better than stable democracy", it's "we cannot have democracy here in Sylvania, because it will degenerate into mob rule followed by a military coup." But this is going to have very different answers in different places and "when would a rational person with perfect information have figured it out" isn't a meaningful question.
I assume that by "European monarchy" you mean 17-18th century absolute monarchy.
Then the guy you are looking for is one François-Marie Arouet, and the time is the 1720's.
Lettres philosophiques or Letters Concerning the English Nation) is a series of essays written by Voltaire based on his experiences living in England between 1726 and 1729 (though from 1707 the country was part of the Kingdom of Great Britain). It was published first in English in 1733 and then in French the following year, where it was seen as an attack on the French system of government and was rapidly suppressed.
( i know that 18th century Britain was far from modern democracy, but of all contemporary countries closest to ideal of modern liberal democracy, certainly compared to France of Louis XV.)
Assume a highly rational actor with as much knowledge of the world as they could realistically have. Roughly what point is the 'turning point' in history after which they should be able to clearly realise that Western democracy is superior to European-style monarchy from a perspective of human welfare?
Clarification: By superior, I mean 'overall superior'- that a variant of Western democracy is a better sort of system to think about when trying to make an ideal system for a country than a European-style monarchy.