This is my personal case and the case of many many Spanish people, as Omicron is sweeping the country.

source?

I would expect that those most at risk from covid are the most likely to get vaccinated for it. I would expect the effect to get smaller once you start controlling for age (which they did not).

Granted, I don't think it's an unreasonable expectation. But I wouldn't state it as being supported by that paper.

'persons who survived a previous infection had lower case rates than persons who were vaccinated alone.'

This _does not mean_ that prior infection confers better protection. The vaccine vs infection was not distributed randomly - inherent risk is a confounding factor. So an equally likely explanation for the conclusion (as opposed to your interpretation of causation) is that the people who are naturally better at fighting off covid (e.g. younger, fewer high-risk factors), are less likely to get the vaccine.