Their usual argument against this would be that these people might've died anyway. How do we know the vaccine increases the odds of death over and above existing causes?

We could try to answer that by looking at the death rate in vaccinated vs unvaccinated groups split by age groups and comorbidities over time, and if deaths tend to clump during the 2 weeks after vaccination in the vaccinated group etc. I highly doubt we'll see that.

Other vaccines have been pulled for far less damage. One good thing is at least people are waking up to the extent of subversion/infiltration that exists in our world.