I don't see any errors in your calculations. I also do not discount the possibility that the vaccination itself might lead to some increase in mortality. In fact, that is exactly what it looks like from the data on the face of it for the period shortly after vaccination - this looks like a very consistent trend too and showing up for a very long period now.

I just cannot believe that there would be such large unaccounted undercounting going on. Especially given that in the new file both death and total counts increased for both the vaccinated and unvaccinated groups. That to me suggests that there is something else going on in the data.

It might be interesting to just try to contact the ONS and ask a direct question about the total population number vs the numbers in this data set - https://www.ons.gov.uk/aboutus/contactus/generalandstatisticalenquiries