Paradigm change in power: from military to economic in Europe

There are a few assumptions here that, in my opinion, muddy the waters.

Let me quickly rephrase.

In the time of Charlemagne, governance was exercised by a king and his military advisers. Currently governance is exercised by a state supported by economic advisers.

First, your premise is false. Charlemagne was supported by non-military advisers - ignoring Alcuin and the Church renders all other analysis meaningless. Second, Charlemagne's court was comprised of economic and military advisers, but at that time there was no distinction between them. Great Lords and churchmen were both military powers and the major landholders. Given that their economy was very primitive, economics was functionally equivalent to holding land. The only way to be a military leader was to be an economic powerhouse and vice versa (you could not muster large levy unless you had lots of land and lots of economic power). Charlemagne's advisers advised him on their best interests which were a combination of economic, military and idiosyncratic.

Second, "economics" undergoes a couple of phase changes in the period. First, the downfall of the ancien regime because the governance of France rejected any and all competent economic advisers. Then Adam Smith articulates real economics for the first time. Before that, economics was idiots in a sandbox with sharp knives. (I am doing a disservice to brilliant men who overthrew the Corn Laws and a dozen other geniuses who happened on the right answer without any of the tools, but I'm writing an H:SE answer, not a book.)

Then John Maynard Keynes articulates macroeconomics for the first time. This is the real inflection point that answers your question - a practical, testable theory of economics that can be used to improve governance.

Third response - which is the second response in a slightly different form. In the beginning of this period, land was wealth and wealth was land. Production was intended for consumption and the concept of "investment" was .... nebulous. Investment meant planting a crop like olives or wine that might not return a harvest for many years. (and was largely the province of the church) Over time the middle class rose and began to contribute value. The middle class is the first class that can invest - they can create tools that make work easier in the future. They can use money to set aside value for the future. But the tools to understand and manipulate wealth are ill understood. At some point the value produced by industry exceeds that produced by land. At that point it is proper and necessary for governance to include economic advice. Somewhere around this point we enter Mercantilism, which is one of the most destructive heresies outside religion. Eventually that transforms into the modern economy.

Washington (who may not have had a title, but was definitely a 1%er) had no clue about economics. The Articles of Confederation managed to absolutely destroy the economy of the colonies. Hamilton barely salvaged an economy from Jefferson (who was quite possibly the worst economist in the history of the United States - he makes Hoover and Coolidge look like geniuses). These are the times that prove that economists are needed for good governance.

Fourth, with a hat tip to @twosheds, there is an evolution to total war that begins (arguably) with the French Revolution, is reinforced by the US civil war and is absolutely clear by WWI - war is no longer won by military leadership, but by the ability of society to transform industrial output to military logistics. This is a book length theme, but twosheds is right - the conclusion is inescapable that economics trumps military today.

Finally your thesis is flawed in that today the Department of Defense, the Department of State and the Department of Homeland Security are far more important advisers than the Department of Commerce. Military advisers still trump economic advice.