Population studies run into different issues you want both. I recall an early study showing vaccination being significantly more effective, but that’s before current variants showed up etc.

Really you can’t draw accurate conclusions from any one study but multiple studies give you a better picture of what’s going on. Large numbers of people falsely believe they had COVID, people who survived COVID are healthier than the general population because the other option is death, the most common infection doesn’t get a strong immune response, vaccine recipients tend to be older and in worse health etc.

That said, an otherwise healthy person who got very sick from COVID and got multiple high quality tests verifying exposure likely does have a significantly better long term protection. But, that’s not representative of the general population that had COVID.