Also, this article provides some arguments based more on biology. https://arstechnica.com/science/2021/10/prior-infection-vs-v...
There is such a thing as testing for neutralizing antibodies by testing the antibodies against the virus directly. But in general, all of the relevant tests seem to be more expensive and harder to standardize.
Every human body creates its own set of antibodies against the virus (or the vaccine). Starting from different strains of the virus, the potential for uncertainty is too great.
Too great for what? Even with 2 jabs, the immunity is <90% against Delta and research shows post-covid immunity is more effective.
Nor is it clear who actually has a "more effective" or even "sufficiently effective" immunity, given that the infection could be up to a year or more ago, and almost nobody knows exactly what strain they got. And no, you can't tell any of that easily, reliably or cheaply by antibody testing.
Proof of vaccination is the cheapest and most reliable proof of a reasonable chance of immunity.