> If the border agent doesn't want you to come into the country, you are fucked.

You are seriously inconvenienced, but assuming your paperwork is in order, you will be allowed into the US. This isn't just against US law, it's a violation of international law to render a person stateless.

This ignores the real point, which is that while you cannot be refused entry to the United States, you can be arrested at the border. ICE these days has mastered the art of making people's detainment so uncomfortable that even those with a right to be in this country end up deciding to leave.