Notwithstanding any fallacies that may exist (can you point out a specific one?), the best available scientific evidence suggests that natural immunity is much stronger than the vaccines against COVID:
"Israelis who were vaccinated were 6.72 times more likely to get infected after the shot than after natural infection". https://archive.is/RlwBc
Here's a very quick summary of what I linked above: In Israel it would be easy to look at the data and conclude that vaccines are providing ~67% efficacy against severe disease/death.
But, once the data is broken down into buckets that help address confounding variables (i.e. different vaccination rates among different age groups), things look very different. All of a sudden efficacy numbers are looking better than 90% for a lot of people.
This will similarly matter a great deal as people try to figure out how long vaccines provide protection. The groups that got vaccinated the earliest in many places were older people and health care workers -- groups which start out at higher risk, and also have a higher probability of less effective immune response to vaccines (older people).
As a result of that, it will be easy for analysts that don't consider that issue to under-estimate the effective time period of vaccines.
The archive.is link you provided isn't working for me at the moment, but to address your statement in the context of the above framework:
The group of people most likely to have been infected with the virus are not the same as the group of people most likely to have antibodies as a result of immunization. In many places, there are a lot more younger people who have gotten infected with the disease than older people. There are other socioeconomic and behavioural differences too.
Given that young people tend to have a more effective immune responses to begin with, and given that they have been shown to have better outcomes after being infected with this virus, it's easy to see a way to incorrectly conclude that stronger immunity results from infection-acquired antibodies, even if the opposite may be true.
In short: Apparent differences may be better explained by the fact that it's a different group of people who have been infected vs those who have not been infected.
We have no idea how complete "known to have recovered" is where as the vaccinated population is well known.
It also doesn't say how "known to have recovered" as well as vaccinated is being calculated.
I'm also not sure what the relevance is given the comparative deadliness of the vaccine and "natural" immunity.
Edit: That's actually another statistic issue. You need to consider the "known to have not recovered." The population of "known to have recovered" is itself a biased selection of those with stronger immunities than the general population.
"Israelis who were vaccinated were 6.72 times more likely to get infected after the shot than after natural infection". https://archive.is/RlwBc