The "bureaucrats in Brussels" are nominated by EU Parliament members, who in turn are elected every five years directly by citizens in European elections. They are there not because the EU is anti-democratic, but because you cannot realistically expect elected representatives to deal with every single detail. Otherwise, you would be in a direct democracy.

Doesn't the same thing happen in the US? Federal and state agencies are full of non-elected employees that take impactful decisions on a daily basis, but I don't see many Americans complaining about a lack of democracy in the USA because of this.

There are many positions in the US that are not elected positions. When's the last time you voted for a supreme court justice?

Bureaucrats make laws in Europe. The Supreme court does not. Big difference.

Actually, the U.S. follows the Common Law system in contrast with the majority of other countries (most of E.U. follows Civil Law). In that system the justice system participates in the shaping of law as it acts as an interpreter of the law and establishes precedent (you might have heard the term case law also [0]). Thus, as you are well aware, law is developed by their decisions and interpretations of the Supreme Court, and other lower courts. For instance, Arizona v. Gant, Roe v. Wade. [0] https://en.wikipedia.org/wiki/Case_law

What you describe is "legal precedents", not laws per se. Sure, that is also a power to make decisions, but it's quite different from enacting new laws altogether.

Legal precedents exist, because laws are interpreted / combined, deduced. Sure, it is not regulatory or statutory law (executive and legislative branches). Yet, courts (and police etc) are bound by the rule of stare decisis. Namely, a court has to abide and be consistent with rulings of higher courts. For instance, if a court says abortion is legal, lower courts, police, and the affected governments (e.g. states, municipalities) have to abide to that ruling/case law.

In other systems, if a law leads to contradiction, or is lacking clarity, the court applies specific 'procedures' to solve the issue (e.g. a law can be ruled unconstitutional or void). Here, the court decides on the details to produce a "reasonable" law.

Edit: But yes, you are right: unlike the legislative branch a court may not propose an arbitrary law (they can advise/make suggestions to the legislative branch).

Are you trying to argue that the Supreme court is just an irrelevant fluff job? Probably also the same argument could be made for general officers and diplomats?