Jumping in for a quick comment on this, because the American and the European are talking past one another across an unrecognized semantic gap. As an American transplant to Europe, it took me a bit to get my own head around it and adjust my terminology.
Generally speaking, for an American, everything that is part of the state apparatus is “government.” The executive, the legislature, the judiciary, the counter at the department of motor vehicles, the administration of the school district … If it’s not private, church, or non profit, it’s “government.” It’s a very broad term meaning those who have been delegated to run the public sector.
But elsewhere, “government” has a much narrower meaning. Think of the political argument raging in the UK right now, which has members of Parliament raging against Liz Truss’s government. The people she has recruited to carry out her political agenda are “in government.” This is distinct from Parliament as the lawmaking body, or from the civil servants who staff the ministries. The ministry’s leader is in government, as they drive the agenda, and the civil servants report to and serve the government, but they are not in and of the government, strictly speaking.
So from these two perspectives: The American looks at the EU’s lawmaking body (accepting the nuance that its laws are not laws per se but must be transcribed into law by member states), and sees people elected and delegated by the public to decide how things are run, so that’s “government.” But the European looks at Brussels and sees only the Parliament and the administrative services supporting it (plus the Bank in Frankfurt and the Court in Luxembourg, etc.), with no centralized executive carrying out a unified political agenda, and does not see “government.”
I hope this clarifies the context of the disagreement.