SCOTUS isn't *supposed* to represent the United States. Wrong branch of government. The US is to be represented by the executive branch, with its people represented by the legislative branch.
That's why the Supreme Court's duties are largely sidelined into writing papers and concluding arguments, setting aside administrative work.
SCOTUS settles what the law says. That may not represent the US, though it would be nice if it did. If you're looking to the Court to represent the United States you're looking in the wrong place.