Why would angels climb a stairway if they have wings?

In the Gospel of John 1:51 our Lord Jesus tells us what the ladder is a picture of:

And he saith unto him, Verily, verily, I say unto you, Hereafter ye shall see heaven open, and the angels of God ascending and descending upon the Son of man.

The ladder is Jesus.

In Jacob's dream the Angels are a picture of communication between Heaven and Earth (downward), and between mankind and God (upward). The Angels ascending up the stairs represent the prayers of men ascending up to God in Heaven; the Angels descending represent God's gracious acts and words towards mankind.

The Angels only go up and down the staircase/ladder because it is only because of Jesus's mediation that God is willing to hear our prayers (receive the angels going Heavenward) and it is only because of Jesus's mediation that God sends gracious words and acts earthward (sends angels down the stairs).

Jesus is the only mediator between God and men (1 Timothy 2:5). God would not be able to communicate with mankind if there had been no Lord Jesus Christ, because our sins have cut us off from him and our sins are such an offence to God. Jesus's life on behalf of sinners and death on behalf of sinners is the only reason God is able to listen to us and to deal graciously with us. Outside of Christ God says to us

your iniquities have separated between you and your God, and your sins have hid his face from you, so that he will not hear. (Isaiah 59:2)

The ladder, with its angels ascending and descending, is a picture of the Jesus's words

"no man comes to the Father except through me" (John 14:6).

Matthew Henry the English puritan, in his commentary, adds another option of interpretation before commenting specifically on why the angels need to use the ladder only:

Jacob saw a ladder which reached from earth to heaven, the angels ascending and descending upon it, and God himself at the head of it. Now this represents two things that are very comfortable to good people at all times, and in all conditions:-

(1.) The providence of God, by which there is a constant correspondance kept up between heaven and earth. The counsels of heaven are executed on earth, and the actions and affairs of this earth are all known in heaven and judged there. Providence does its work gradually, and by steps. Angels are employed as ministering spirits, to serve all the purposes and designs of Providence, and the wisdom of God is at the upper end of the ladder, directing all the motions of the second causes to the glory of the first Cause.

The angels are active spirits, continually ascending and descending; they rest not, day nor night, from service, according to the posts assigned to them. They ascend, to give account of what they have done, and to receive orders; and then descend, to execute the orders they have received.

Thus we ourselves should always abound in the work of the Lord, that we may do it as the angels do it (Psalm 103:20,21).

This vision gave very seasonable comfort to Jacob, letting him know that he had both a good guide and a good guard, in his going out and coming in, - that, though he was made to wander from his father's house, yet he still was the care of a kind Providence, and the charge of the holy angels. ... Jacob was now the type and representative of the whole church, with the guardianship of which the angels are entrusted.

(2.) The mediation of Christ. He is the ladder, the foot on earth in his human nature, the top in heaven in his divine nature: or the former in his humiliation, the latter in his exaltation. All the communication between heaven and earth, since the fall, is by this ladder. Christ is the way; all God's favours come to us, and all our services go to him, by Christ.

If God dwell with us, and we with him, it is by Christ. We have no way of getting to heaven, but by this ladder; if we climb up any other way we are thieves and robbers. To this vision our Saviour alludes when he speaks of the angels of God ascending and descending upon the Son of man (John 1:51); for the kind offices the angels do us, and the benefits we receive by their ministration, are all owing to Christ, who has reconciled things on earth and things in heaven (Colossians 1:20), and made them all meet in himself (Ephesians 1:10).