"Vaccine effectiveness evaluated at least 7 days after receipt of the third dose, compared with receiving only two doses at least 5 months ago".

Literally no control for recency. Newer research (admittedly, not yet peer-reviewed) seems to show that efficacy of the vaccines dwindles over time. As little as 90 days. [1]

The results from the booster may have more to do with having received a vaccine more recently than they do with the overall efficacy of any of the vaccines.

https://www.medrxiv.org/content/10.1101/2021.12.20.21267966v...