Nothing in his analysis is right. First, he uses that infection stats for a variant (Omicron), but the vaccine targeted the origina known to be less effective against it. So technically speaking, assuming that most unvaccinated people were infected (and some died) with other variants by then, they were better prepared against Omicron. Not making and distributing an Omicron-specific vaccine was a logistics issue, not a technical one.
That being said, remember that Omicron turned out to be a relatively less lethal variant, so being vaccinated was a net win, because a percentage of anti-vaccine people died with the original virus, and the Delta variant while trying to gain natural immunisation against it. We can't compare the risks of vaccination and actually getting sick.
Second, he is comparing case numbers (not deaths, or hospitalisations), which won't work, because it only demonstrates that vaccinated people are more likely to go and get tested when they feel sick, and unvaccinated people are less likely as anti-vaccine community (I know that not all unvaccinated people are anti-vaccine) generally don't think that they have a responsibility to keep other people safe.
Citation needed.