I disagree and here's why. 1 Peter 3:18-20 - "Christ also suffered. He died once for the sins of all us guilty sinners although he himself was innocent of any sin at any time, that he might bring us safely home to God. But though his body died, his spirit lived on, 19 and it was in the spirit that he visited the spirits in prison and preached to them— 20 spirits of those who, long before in the days of Noah, had refused to listen to God, though he waited patiently for them while Noah was building the ark. Yet only eight persons were saved from drowning in that terrible flood."
From Adam on, the only deciding factor on what and who is acceptable by God only depends on their faith. If you think all the heroes of faith like those listed in Heb. 11 aren't going to be in heaven with us, you are sadly mistaken. Their spirits are there now awaiting the resurrection where they will be given the same type bodies we are according to 1 Cor. 15:51-52.
You said, "Immortal beings don't need to eat to keep living." The bible seems to say that even after we have put on our "immortal" bodies we will still need to partake of the tree of life via eating.
You said, "They can't be put to death because immortality means deathlessness, indestructible." There are many Christians who believe that as we ae given eternal life, the souls of the unsaved will burn in hell forever, and others say there will come a time after the GWTJ that death will be destroyed and all those eternal souls will die. Only the creator of those souls has the right and power to make that determination, in other words if he can make our mortal bodies immortal, then surely he can do it in reverse.