No, this isn't the verse, but I think it brings understanding to that verse.
This isn't the branch that is not in Jesus, this is a branch that is in Jesus, but does not bear fruit. The Father, the husbandman, "carries" that branch away, "airei", to take up and carry away.
There are branches that are in Jesus, and those that are not.
John 15
1 I am the true vine, and my Father is the husbandman.
2 Every branch in me that beareth not fruit he taketh away: and every branch that beareth fruit, he purgeth it, that it may bring forth more fruit.
3 Now ye are clean through the word which I have spoken unto you.
4 Abide in me, and I in you. As the branch cannot bear fruit of itself, except it abide in the vine; no more can ye, except ye abide in me.
5 I am the vine, ye are the branches: He that abideth in me, and I in him, the same bringeth forth much fruit: for without me ye can do nothing.
6 If a man abide not in me, he is cast forth as a branch, and is withered; and men gather them, and cast them into the fire, and they are burned.
There are three. The branch in Christ that does not bear fruit, the Father carries away. The branch in Christ that does bear fruit, the Father purges so it will bear more fruit. And the branch not in Christ is gathered to be burned.
What is the branch that is in Christ, but does not bear fruit? What does it mean that the Father carries this one away, but the branch not in Christ is burned? What is the difference between these?
I understand this to mean that the branch that is actually in Christ, born again, but unfruitful, they may find themselves carried off by the Father, in an untimely death. The born again person who is fruitful the Father will work with to become more fruitful. And those who are not in Christ, not born again, these are gathered and burned.
With this in mind, saving someone from death could be speaking of physical death, not spiritual death, as that person has become fruitful again, showing this to be the purging work.
Much love!
"What is the branch that is in Christ, but does not bear fruit? What does it mean that the Father carries this one away, but the branch not in Christ is burned? What is the difference between these?"
I would say that the branch that is in Christ but does not bear fruit could relate to the meaning of "ariei" (to take up).
If we take this in a literal sense...what would the caretaker of a vineyard do when a branch is weak and not bearing fruit? He would lift it from the ground and attach it back to the vine to get life? Don't they do that IRL?