What’s happening is that there is no central database of all US citizens, but there is information on every lawful immigrant.
I think you’re trying to say: “they have no qualms deporting x, and x has a stronger claim to being here than y, so clearly they’d have no qualms deporting y”.
The reality is completely different. If you’re a lawful immigrant, they have your fingerprints, photograph, travel history, address, and a whole lot of other information already, and you can use that to prove your identity. Of course, you may then be subject to a fine for not carrying your documents, as all aliens are required to pursuant to INA 264(e) and other provisions.