Why to Burn Withered Branches.

John 15:5-6

5 • I am the vine; you are the branches.

• Whoever abides in me and I in him,
he it is that bears much fruit,

• for apart from me you can do nothing.
6 If anyone does not abide in me
he is thrown away like a branch and withers;
and the branches are gathered,
thrown into the fire,
and burned.

If a vine branch is grafted into a vine successfully then it bears grapes as its harvest. The vine acts as a source of life for the branch and, although it is the branch that bears the grapes, without the vine, the branch could produce nothing. Using the metaphor of the vine and its branches to describe the Christian life is fairly straightforward because we have a ready understanding of how God's grace flows to us from Jesus Christ and how our fruitfulness in all these graces such as love, joy, peace, etc. [Galatians 5:22] are found in us because we're found in him.
What might not be so straightforward is why ungrafted branches are to be mentioned at all but the model needs the added explanation to reinforce its point that believers ought to be 'in Christ.' It isn't looking like a branch that makes the branch useful but it's being grafted into the vine and a branch that isn't grafted into the vine is worse than useless. It isn't for fruitlessness that the withered branches are gathered up and burned, nor is it for the tidiness of the vineyard alone. Some genius might come up with a use for vine branches that wasn't applicable 2000 years ago but the picture then was of the primary uselessness of the vine except for growing grapes and its secondary worse-than-uselessness of piles of withered branches harbouring disease.