Or they just move their "headquarters" and the US part of the business will be a subsidiary.

This is an old, and well tested strategy.

E.g. Commodore International formally had its head office in The Bahamas, but the entire leadership team worked out of the US.

You can try putting more constraints on what will get a company considerd a US company to catch those kinds of structures, but as you indirectly point out, there are really only downsides to playing that game.