First, yes, AFAIK herd immunity may very well require true immunity. Although I'm also thinking about the possibility of reducing infections by enough that the spread is just too slow to get through our quarantine strategies. In that case, vaccines don't need to provide full immunity.

Second, these vaccines are not and will not ever cause more deaths than covid. I'm not sure what you mean by looking at all causes of death.

You can have a population where infections happen with no vaccines and get herd immunity. You can use vaccines we have now, infections still happen, and get herd immunity from the natural immunity from infections except that fewer people will be severely affected and dead because we know the vaccines make death and serious illness less likely. It's wild that you think somehow herd immunity is impossible in a world with these vaccines, but it is inevitable in a world where these vaccines don't exist.