The text you use as your source certainly shows that death is the result of sin. Building on that source, another text gives further information relevant to that fact. It is in Romans chapter 6, where Christians are told that they have been set free from slavery to sin, the result being holiness, and eternal life.
This, however, does not mean that they avoid physical death because they still were, and still are sinners. The accumulated body of sinful works they did in their life still means that they have to die, because "the soul that sins shall die". Further, the last verse in this chapter adds a vital fact. It is that:
"The wages of sin is death, but the gift of God is eternal life in
Christ Jesus our Lord." Romans 6:23
Now, a wage is earned, is it not? Work is done, and payment is an obligation; the worker must be paid for the work done. Thus Romans 4:4 confirms that:
"Now when a man works, his wages are not credited to him as a gift,
but as an obligation."
That is why God pays every sinner the wage they have earned for the body of sinful works they have built up. They receive death. But nobody can receive their wages then try to hand it back to the one who paid them in order not to receive the wage! That is absurd! God's wage to every sinner is physical death and nobody can refuse to receive it. They must die. They will die. They do die. Ezekiel 18:4 is in perfect harmony with Romans 6:23.
All of this is contrasted with the amazing gift of forgiveness of sins which results in everlasting life, to those who believe. Romans 4:5 adds that to the man who does not work but trusts God who justifies the wicked, his faith is credited as righteousness (then Paul quotes Psalm 32:1,2 and refers to the faith of Abram). This means that sinners must stop working to try to "earn" God's forgiveness because they can only earn death, due to everything they do being polluted with sin. Once they believe that only the finished work of Christ at Golgotha can enable the free gift of forgiveness and everlasting life, they are set free from their slavery to sin. They still receive their wage of physical death, but even before they receive it, they have received God's gift of salvation and step instantly from death to life, the moment they believe.
This explanation shows why nobody can 'atone' for their own sins. Dying is the result of sin and does nothing to remove that sin because death is proof that this body of sinful works has been done.
On the other hand, the only man who never sinned, voluntarily gave himself over to physical death in order to atone for those who would repentantly believe in what he had done. Jesus need never have died, for he had never sinned. But he came to give his perfect life an offering for sin - Romans 8:3. That is why death could not hold him, for death can only claim sinners. Christ arose, triumphant over sin, death and the grave. All who trust in that provision, in faith, are baptized into Christ's death, and raised in newness of life.
You ask if God's method of settling the issue of sin means that Jesus died for nothing? On the contrary, God's method of settling the issue of sin meant that Jesus had to die as the only perfect sacrifice for sin there has ever been because our deaths simply confirm that we are sinners! Only Jesus' death can take away our sins in God's eyes. That is why, when we put our faith in that, the issue of our sin is settled.