For vaccines for which the vaccine has few side effects, this is, AFAICT, not usually considered worthwhile — providers seem to prefer giving an extra MMR dose to people with unknown vaccination status for example. For COVID, at least with current vaccines, maybe it would make sense.
(ISTM, based on available data, people who have had COVID ought not to be required to get two doses, but one might be reasonable from a public health perspective.)
How do you make sure people aren't faking vaccination status, as you say? A positive PCR test on public record is in fact more trustworthy.
The tests used in most places have significant false positives/negatives on a population level. You may think you were infected when you weren't. With vaccination you know whether you were vaccinated.
(My son had a bout of severe pneumonia around the same time. We wondered, too. Not COVID, it turns out.)
Therein lies a significant problem with "I don't need the vaccine, I have natural immunity!"
That's just as trustworthy. Probably more since what it asserts is less variable than how long you maintain antibodies.
The link I provided "is an electronic vaccination record drawn from the data stored in the California immunization registry."
It doesn't accept user submissions; it only accepts data from recognized health organizations. The website allows you to put in your name and birthdate and get a QR code that can be scanned and will take you back to the website, showing dates you were vaccinated. It requires someone confirming that the QR code is directing them to the correct website, and confirm name and birthdate, but that's far more available to actually check than a PCR test.
Meaning no need to rely on easily faked vaccination cards; you instead ask for a QR code, scan it, and confirm it took you to the government site, with a name and birthdate that matches some government issued picture ID. Still 'beatable', if you have good fake ID for a person who has been vaccinated, but just as good in 'security' as relying on a PCR test (since it's still ultimately pulling from a publicly readable, only privately writeable DB), and probably a better assertion than "PCR test", since it can work for everyone, and tells you the date of vaccination (whereas date of antibodies being detected doesn't tell you anything about whether they still have natural immunity now).
So, you can't prove a negative, that being that someone has not been vaccinated. But you can prove that you had covid.
CA is larger than LA county, so immediately is more effective than what you proposed in your OP.
There are also multiple initiatives to make country-wide databases of vaccination status available via app. Test status? False positives, false negatives, can be done at home, etc, and -still only applies to people who have had COVID-. Everyone without a health issue (that should keep them out of public spaces anyway) can get vaccinated.
1. You cannot prove someone has been vaccinated as the system currently exists. That ship has already sailed. Anyone can opt out by lying that they got vaccinated somewhere that doesn't keep public records. Those mandating vaccinations have have to accept this because vaccines aren't approved for more than 2 (or 3) shots, so you can't force people to get vaccinated again if they don't have a verifiable record. Even if there's a worldwide database rolled out, it's too late, too many people got vaccinated off the record.
2. You CAN prove you've had covid with a high degree of certainty. Recent tests have low false positive rates.
Why let people slip out of the vaccine mandate with an easy lie yet corner those who have already had covid and therefore better immunity than the vaccinated? It's nonsense that ignores reality.
You do the society a favor, and the app they provide lets you know if you continue to have covid19 antibodies.