I see it as God alone has immortality and can give life to who ever He chooses.
having life and the ability to create life is different. Having the gift of life every day and never coming to death is what we will have because God gives us life.... We do not have our own self existent source of life, we cannot live without Gods life in us. while we are connected to the vine we have life, sin cuts us off from the source of life and we slowly wither and die.
Without God can anything exist and live?
Before the fall, Adam had the breath of God in him and no sin to separate him from God.
I can't see why I should assume he was in a perpetual state of death that needed to be temporarily 'fixed' every other day by eating from the tree of life.
So if he is sinless, that makes sense to me as equivalently deathless. God is with him and sustaining him, until by sin he becomes separated.