I think there are a couple points you missed: From that Table 5. in the PHE report, younger ages 20-59 are having less deaths still than unvaccinated. It is a clear pattern from this table the younger you are - the better off when double vaccinated.

68.1% efficacy against death is huge, and for that period is likely to be mostly delta variant, we can't just forget about Alpha variant which vaccine does amazing with, which the previous data shows.. not sure how this points to them being "government propaganda" campaign. Although I agree it's a bit disappointing they are not painting the full picture.

But people reading that "73% of new deaths were unvaccinated" would cause panic among people that don't understand you need to take into account ~90% of population aged >60 have been double vaccinated for a while now .