I rather think the fiction is that they serve the government, but I also have The Creature from Jekyll Island in the backseat for light reading so my perspective probably is heavily biased.

You aren't biased at all [1]. Our dear colleagues (along with the majority of the American populace) are just deeply ideologically committed to the idea of the Federal Reserve being a public institution, because they've been told this implicitly all their lives.

The truth, which is difficult to uncover and requires digging into some legislation, is of course unsettling, because it points to the nature of the American system and which institutions have power. Hint: if an institution is immune to bankruptcy or prosecution, it's probably the one in charge.

1. https://publicbankinginstitute.org/money-banking-basics/