I recently became much more pro-total-unification, so let me give you this counterpoint: individually, European nations are no match to the major superpowers, neither economically nor militarily. We'll get gutted by divide-and-conquer approach. In contrast, bound much closer together (particularly with some form of pan-european armed forces), the EU would become a proper global superpower and a counterbalance for the USA and China.

Even more centralized than the USA? That must be an outsider's view. You are aware that each state has its own tax laws? Corporate laws? Even criminal laws and traffic laws? And so on? Obviously there is federal law, too, but that's more of an umbrella covering all states in addition to local laws. The USA is a federation, just like Germany, Switzerland, or Austria are federations, and one common complaint you get out of these nations all the time is that they are not centralized enough!

The EU, on the other hand, is not a federation, it is a association of nations trying to figure out how to go together, be it as a federation or something different, because in today's world you need size and power to survive. Any EU state alone would be insignificant on the world stage and being sidelined one way or the other. The most powerful inside the EU would suffer the most, e.g. Germany or France, because while the EU is second or third on many metrics, the individual nations are small. Depending on which ranking you look at, the EU is just behind the USA in military and economic power, sometimes it is behind China in economy, too.

The strongest individual economic power of the EU is Germany, which globally usually comes in 4th place after the USA, China, and the EU as a whole. But this is deceiving, when you look at the absolute numbers. Germany's output is impressive for its size, but is still only about a quarter of China's! Germany alone is a dwarf and without the EU would be inconsequential. The post-Brexit UK is just learning this the hard way and all the sovereignity does not matter a damn. The top leadership of EU nations are not stupid, even if they could explain their reasoning better, and thus try to keep the EU going, despite its many flaws.

This is a Chesterton's Fence situation.

It seems that in your worldview value and wealth does not increase, just changes hands.

This calculation is flawed. German state collects much more than 5€, because there is VAT etc and, of course, employees of that company pay income tax from their share of that 100€ they just got back.