How does immigration work without knowing who your citizens are? How would you prevent a non-citizen from getting a passport without having such a record?

If you want to get a passport the onus is on you to provide sufficient documents to prove citizenship. Because the US has birthright citizenship this is generally as simple as proving that you were born in the US or that either of your parents was a US citizen.

On a case-by-case basis it's generally pretty straightforward to determine whether someone is a citizen, but there isn't a list. The closest thing is probably the social security administration's register of people who have applied for a social security number, but even that doesn't cover everyone.