There should be enough data to use recovery at least to some extent since it seems quite similar to vaccination to the laymen. We don't know how long vaccination will last yet we are willing to accept that even for those who were vaccinated in November so why shouldn't we accept recovery for something too
Again, I feel like you're conflating two separate issues - immunity and policy.
Re:immunity, we just don't know yet what factors are responsible for one person having a breakthrough infection or reinfection, while another person remains immune. Our inability to answer this question isn't criminal... there just haven't been enough cases with enough data collected from them.
Re: policy, I don't know what NYC will decide regarding whether to accept an antibody test as "proof" of immunity. It's not scientifically valid proof, but perhaps it's good-enough-proof for the purpose of allowing admission to public venues.
The numbers I gave before were for the US in general, where recoverees make up about 10% of the population. And since many recoverees have vaccinated, the unvaxed recoverees are, let's say, 5%, so easy to ignore. But in NYC, the covid cases were higher than elsewhere, so the recoverees are a larger percentage of the total - I don't know the numbers - and I would assume that the NYC authorities will have to come up with some solution.