No, the death rate in the ONS data from England is age adjusted. So among the same aged cohorts, the vaccinated are more likely to die than the unvaccinated. Almost twice as likely for the last 6 months straight.
is it also standardized for immune system status? Diabetes? Etc etc etc
The unvaccinated population in that age group is significantly younger than the vaccinated population. As you would expect, older people tend to die more often.
This is also mentioned in a footnote in the ONS data (and this is why they tend to focus on age-standardised figures).
Is the average covid death rate age adjusted?
This kind of misconception is so common it even has its own name: Simpson's paradox. It's laughable to think that the vaccinated population, compared to unvaccinated, is dying at the rate suggested by these numbers and no one is ringing the alarm.