Understood. But just as the first death wasn’t final, neither is the second death final.
Christ’s description of the rich man’s torment in hell is further evidence for punishment.
You present a false dichotomy regarding the wicked. God will remember them regardless whether He annihilates them or eternally tortures them. It’s the saved who won’t remember them. God will either expunge them from our memory, or He’ll change our thinking such that we praise His justice. That’s the only ways He can achieve this:
Rev 21:4 And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away.
This promise is to those in heaven (verse 3). It of course doesn’t apply to the lake of fire. Your argument falls apart when you consider Satan, the Beast and the False Prophet are in the lake of fire too. And all of them are in the most excruciating pain imaginable for eternity.
Then the Bible is false by such reckoning, for if the Second Death is not final - then there is no need for a Savior, because life exists whether we bow to Him or not.
I do not subscribe to the tradition, whereby it is impossible to die because all live forever. Does scripture really teach life continues for eternity regardless of whether you are wicked or righteous? Only the place you go differs? That does not comport with my understanding of scripture.
The wages of sin is DEATH according to the bible. We've all earned death. That is why we need a Savior. It does not state that the wages of sin are eternal life in hellfire for those who are wicked. Unless you chose to redefine what death is as some do, i.e.; eternal separation from God. Yet the bible tells us that there is no consciousness or knowledge when dead. Just cessation from existing.
In any case, it makes no logical sense for the saints to await a resurrection if there is no need for one. We live forever according to tradition - only the place and the well being of the individual changes; one to eternal bliss and one to eternal torment and suffering.
I cannot square that tradition with the scriptures.