Correct. And that's the original intent of the TSA website/account. Flight schools (especially in FL) take it very seriously. I can even understand the TSA/FAA wanting all active CFIs to have an account. Overkill, but understandable.
The issue comes into keep documentation. What is an independent CFI supposed to do? Keep the birth certificate? That's kind of a PII nightmare. Make a note in the CFI's files like, "Certified Mr. number9 is a U.S. citizen on 11/6/24?" Sure, but the CFI could make that note at any time later on.
No one is going to care about the documentation unless something bad happens, and even if that happens, verifying the citizenship won't be the actual cause of the incident since if the person is already a citizen, there's not requirement to enter them into the TSA system.