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.
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.