Well, first let me just try linking you to some studies that I think are reasonably sound, and you can peruse them and see if you think the methods are sound. And maybe you can dig into the CDC's favorite study and look at their methodology a bit. If that's not enough, I'm happy to go through more details about the dodgy CDC methodology and questionable analysis and statistics (hard to check--neither code nor data is available), and the comparatively fairly sound methodology and statistics in some of the others.

Also, note that I'm not claiming that full vaccination (or boosting) isn't an improvement over infection-mediated immunity, just that infection-mediated immunity is on par with the best vaccines. (So Europe is being rational, for example, when its vaccine passports require vaccines or recovery from infection.)

Anyway, here are some papers.

Qatar--carefully matched samples out of a potential pool of many thousands, showing 90%ish protection on the same strains that mRNA vaccines got 90%ish protection on when studied: https://www.medrxiv.org/content/10.1101/2022.01.05.22268782v1

Israel--matched vaccinated people against people who caught COVID-19, showing substantially better protection from previous (fairly recent) infection (but without controlling for which strain was involved, which isn't entirely fair): https://www.medrxiv.org/content/10.1101/2021.08.24.21262415v1

UK--not a paper, but public health identifying possible reinfections and coming up with a very low number (~0.4%), considerably better than any reports for vaccines...not very well controlled, but nationwide so a huge sample size: https://www.gov.uk/government/news/new-national-surveillance-of-possible-covid-19-reinfection-published-by-phe

More papers exist, but I think these, from three different countries, using three different methodologies, all coming to the same kind of conclusion (same or better) should serve to make the point. (If you want to track the relative rates of waning of vaccine-induced and infection-induced immunity, you need yet more papers, but again, the infection looks equal or better from what I've seen, though the data isn't as clean.)

Let me know if you'd like me to go through more of the particulars that show why the CDC's favorite study has more serious methodological flaws than the others, so even head-to-head against a single paper you should doubt the CDC's.