Well, The Justice Department at least can't do that, because they can't write laws.

Since when does “the US” consist exclusively of “the Justice Department” and not, e.g., the FTC (which writes antitrust regulations within existing law) and Congress (which writes laws).

I honestly don't know much about this, but isn't it standard for government lawsuits to be "United States vs John Smith"?

To the extent that is correct [0], that doesn't justify reinterpreting “the US should adopt regulations like X” to mean “the Justice Department and not any other part of the US government should adopt regulations like X”.

[0] Its essentially those actions where the DoJ is the agency representing the government interest, including all federal criminal cases and some federal civil cases. Civil cases by other agencies have the agency name; so the antitrust complaint by the DoJ, 15 state governments, and the District of Columbia is United States of America, et al. v. Apple, but the SEC action against Coinbase was Securities and Exchange Commission v. Coinbase, Inc., and Coinbase Global, Inc.

If only US had a sort of a legislative body where you elect people to and then they actually can write laws? That would be great.

The EU legislation wouldn’t fly for a second in the US system of law.