This statement was the source of widespread criticism and I in fact criticized it myself in the writeup.
I considered going for a deeper debunking but held off for the initial draft. I might circle back if people feel it's valuable. Let me take a quick stab though:
When we talk about immunity we need to realize the term is used imprecisely by laypeople (btw I'm totally a layman if it's not already clear).
I think of it simplistically as two components:
(1) The presence of actively circulating antibodies in the bloodstream. This is what the (oddly controversial) serology studies are measuring. It is thought that having a significant quantity of these antibodies prevents infection - i.e., what most people envision when they talk about immunity.
(2) Even after the antibodies have faded, there are still Memory B Cells, which lay dormant up to decades, waiting for exposure to the characteristic antigen (in this case, an antigen telling them that they have encountered SARS-CoV-2), at which point they resume and rapidly scale up production of antibodies.
The thinking here is that reinfection is likely possible after a sufficient length of time - whether that's a couple months or a couple years isn't yet known - but when you do get infected, your immune system will respond sooner, more strongly, and thus you will achieve a far lower peak viral load meaning a less serious infection with reduced transmisibility.
This is a robust mechanism that we see across tons of disease, including common cold coronaviruses. In my completely uncredentialed opinion, this effect is so common and well-supported that we should essentially assume it happens until we really have proof that it doesn't. I know that might sound backwards, but it really is such an enduring mechanism.
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Here's another argument you might prefer. We know that at least hundreds of thousands of people have successfully recovered from COVID-19. Loosely, we can divide the immune system into the "non-specific" and the "adaptive" subsystems. Given that we have seen extensive recovery from this illness and that those who have recovered have detectable levels of antibodies, and furthermore that there is a case of a women who could not manufacture antibodies due to a rare auto-immune disorder who failed to recover, it stands to reason that the same mechanism that provides medium-term immunity is the mechanism responsible for recovery.
That is to say, that it's very farfetched to think that people are recovering purely from non-specific immune responses and that the detectable presence of circulating antibodies is just a red herring. But I suppose you could try arguing that if you wished.