I can present to you a very good reason not to use immunity status. Every idiot and their brother was discussing how they had already had it when statistically this was absolutely impossible. Everyone who had the sniffles from late 2019! to current is absolutely sure they have had it because it would be more convenient to be immune than not.

It would be impossible to give antibody tests to 60M antivaxxers in any reasonable time frame. Instead of asking people to wait in line a year for a test that has very little medical use the politically expedient thing would be to ask for the equivalent of a doctors note saying they pinky swear their patient had covid. Some states/some doctors would fire up the presses and within a month 95% of the antivaxxers there would have a certificate regardless of actual status.

Since vaccination is actually in everyone's interests even if they have already had covid this would lead to less people being vaccinated and it would be against even the anti vaxxers interests and furthermore locations that only allow vaccinated or tested people would now be equally full of infectious people as vulnerable people with a newly false sense of security.