The "national" level will always matter. It's a problem of governing and risk management. Different regions have different problems, and the bigger an organisational structure, the more complicated. The hierarchical system with different levels of legislation and regulation at least works.

In some sense we are already working towards a global government by claiming human rights to be universal. Out of that endeavour there will (in the far future) emerge a central authority or assembly which effectively enforces human rights and free trade.

Would it not be possible to change the standard US/THEM mental model without meeting a new THEM?

Imagine having to explain that you represent some people only because you told those people you represent them. What with all the independence movements, the direction of the world seems to be toward a million nations, not world government.

There are trends in both directions, toward government at levels beyond traditional nation-states (this is particularly visible in European institutions and things like the International Criminal Court, a permanent international court with jurisdiction over persons rather than nations), and toward smaller nation-states. Its not hard to imagine an end state where a narrow range of basic functions were handled by a world government (or a collection of function-specific, separately-constituted, global institutions), but where the unit below that was a set of governments (which might be characterized as "national") over regions smaller than today's nation-states (at least, the medium-to-large nation-states of today.)

This is what I hope for, though I'd like to think it happens even without aliens.

Maybe the hypothetical aliens also only represent their own nation?