>We don't really have a standard way to definitively say "I am a citizen" in the US.

In most countries of the world, the best way to prove your citizenship is to apply for a visa. That is you world apply for a US visa and get an official rejection, because US citizens don't need/cannot get a visa, and the rejection document would be the proof of citizenship.

.. that seems extremely dangerous, because I wouldn't trust that refusal to not raise red flags for the rest of your life. I've not heard of people routinely doing this or announcing it as a valid method of proof of citizenship which they accept.