to cut off or cut down: to sever something from what it grew on or belonged to, and be rid of it
This verb joins the ordinary word for cutting to a prefix meaning "out of," and that is exactly what it pictures: something severed from the place it was attached to and then discarded. All ten New Testament uses keep that picture. What changes is only what gets cut. A tree is felled at the root, a hand is taken off and thrown away, a branch is cut out of the trunk it fed from, and in one case an opening someone was reaching for is taken away.
The uses hold together because the cut is always meant to be final and always costs something that was still living or still useful. The tree does not grow again where it stood. The hand is gone for good. The branch no longer draws anything from the root. Even the one use with no wood or flesh in view works the same way: what is removed is removed for good, not merely blocked for now.
In the Gospels this is the standard image for God's judgment on fruitlessness. John the Baptist says the axe is already lying at the root, and the fig tree in the vineyard gets one more year before the order is carried out. Paul then keeps the same picture in Romans 11, where Gentile believers are warned that they too can be cut off from the tree they were grafted into. Because English needs a different verb for each object (a tree is cut down, a hand cut off, a branch cut out), the thread running through these passages is easy to miss in translation, though the Greek uses one word throughout.
The cutting is severing, not pulling up: a plant torn out of the ground root and all takes a different verb (ἐκριζόω). The axe in Matthew 3:10 lies at the root, but what it does is go through the trunk.