I like to joke that I have more documenting showing my legal right to be in the US than my wife's grandmother. My wife's grandmother has a birth certificate but it is water damaged, illegible, and falling apart. Besides that she has no state ID (driver's license expired twenty years ago), and few other forms of identity.
Americans seem quite dead set against a national ID scheme (and associated database). The problem with that is that we wind up with a de-facto one, SSNs and social security cards being your main proof of who you are and your legal status. Birth certificates are problematic because they've never been standardised, and have no real fraud mitigation mechanisms (e.g. watermarks). Certainly my daughter's US birth certificate would be trivial to duplicate.
This is in no small part why I ordered my daughter a US passport even before it was needed. It is one of the best forms of ID and citizenship proof you can have, even if it expires you remain in the database indefinitely. It is also why I ordered myself a US passport-card, so I have proof of citizenship in my wallet at all times.
ICE, interior immigration checkpoints, and other initiatives seem to be on the rise. Even legal immigrants and US citizens need to protect yourselves, particularly if you're more likely to be profiled due to race.