Public health agencies are in agreement that the vaccines provide additional levels of protection even for those who have had COVID:
"You should get a COVID-19 vaccine even if you already had COVID-19.
Getting sick with COVID-19 offers some protection from future illness with COVID-19, sometimes called “natural immunity.” The level of protection people get from havi ...[text shortened]... ated if you wish; you just can't travel to Australia unvaccinated without a valid medical exemption.
Here is an excerpt from the link below:
"The CDC study did not create a cohort of people to follow over time. Instead, they identified people hospitalized with Covid-like symptoms, and then they evaluated how many of them tested positive versus negative for Covid. Among the vaccinated, 5% tested positive, while it was 9% among the Covid recovered. What does this mean?
Though the authors do not mention it, they adopt a de facto case-control design. While not as strong as a cohort study, this is a well-established epidemiological design. The first study to show that smoking increases the risk of lung cancer used a case-control design. They compared hospitalized patients with lung cancer and found more smokers in that group compared to non-cancer patients, who served as controls. Note that if they had restricted the control group to people with (say) heart attacks, they would have answered a different question: whether smoking is a larger risk factor for lung cancer than it is for heart attacks. Since smoking is a risk factor for both diseases, such a risk estimate would differ from the one they found.
In the CDC study on Covid immunity, the cases are those patients hospitalized for Covid disease, having both Covid-like symptoms and a positive test. That is appropriate. The controls should constitute a representative sample from the population from which the Covid patients came. Unfortunately, that is not the case since Covid-negative people with Covid-like symptoms, such as pneumonia, tend to be older and frailer with comorbidities. They are also more likely to be vaccinated.
Suppose we wanted to know whether the vaccine rollout successfully reached not only the old but also frail people with comorbidities. In that case, we could conduct an age-adjusted cohort study to determine if the vaccinated were more likely to be hospitalized for non-Covid respiratory problems such as pneumonia. That would be an interesting study to do.
The problem is that the CDC study answers neither the direct question of whether vaccination or Covid recovery is better at decreasing the risk of subsequent Covid disease, nor whether the vaccine rollout successfully reached the frail. Instead, it asks which of these two has the greater effect size. It answers whether vaccination or Covid recovery is more related to Covid hospitalization or if it is more related to other respiratory type hospitalizations."