What are typical Christian responses to The Grand Inquisitor?

Opening

The Church does not base her theology on rebutting, as in this case, the arguments advanced by the Grand Inquisitor parable.

The Church's theology delves into Divine Revelation = Sacred Scripture + [Holy] Tradition.

Nevertheless, the Church's theology can rebut those kinds of arguments. In line with the criteria for answers on this site, this post starts from the theology of the Church and not from the Grand Inquisitor parable's position.

The following is a Catholic understanding of the Temptation of Jesus in Mt 4:1-11. It is a commentary by members of the Faculty of Theology of the University of Navarre on the passage in The Navarre Bible New Testament Compact Edition.

Matthew uses this episode of the temptations in the wilderness to depict Jesus as the new Israel, in contrast to the old. Jesus is tempted, as Moses and the chosen people were in their forty-year pilgrimage in the wilderness. The Israelites yielded to temptation: they complained against God on account of their hunger (Ex 16:1ff), demanded a miracle when they had no water (Ex 17:1-7), and adored the golden calf (Ex 32). But Jesus does not yield and , by overcoming the temptation, "reveals the way in which the Son of Man is Messiah, contrary to the way Satan proposes to him and the way men wish to attribute to him (cf. Mt 16:21-23)" [CCC, 540].

With such an understanding, I hope the confusion you have as to why the Grand Inquisitor is considered to be such a powerful argument for atheism and against theodicy will begin clearing up.

Please see also: Home > Summa Theologica > Third Part > Question 41. Christ's temptation | New Advent.

The chapter itself seems to state that Jesus set the bar much too high for man with respect to salvation, but isn't it true that humans were never intended to reach the level of holiness that Jesus attained and that salvation is a gift to those who repent?

If we may have been unable to succeed before, with the coming of the LORD and his help, we can now overcome temptation and sin.

From CCC, 540 referenced above:

This is why Christ vanquished the Tempter for us: "For we have not a high priest who is unable to sympathize with our weaknesses, but one who in every respect has been tested as we are, yet without sinning."[Heb 4:15].

And

1 Jn 5:3-5(RSVCE)
3 For this is the love of God, that we keep his commandments. And his commandments are not burdensome. 4 For whatever is born of God overcomes the world; and this is the victory that overcomes the world, our faith. 5 Who is it that overcomes the world but he who believes that Jesus is the Son of God?

[B]ut isn't it true that humans were never intended to reach the level of holiness that Jesus attained[?]

That is precisely our calling:

Mt 5:48 (RSVCE)
48 You, therefore, must be perfect, as your heavenly Father is perfect.

Why is Ivan's description of the suffering of children such powerful ammunition for rejecting God?

Why is there evil? Why does God permit suffering? Why do the innocent, like children, suffer?

We know that these are very deep questions whichever way they are approached, theologically, philosophically, etc.

We only begin to understand them, and only just, after the coming of the LORD Jesus Christ, the all-benevolent, all-good, all-holy, all-powerful God. He himself suffered.

cf. Gaudium et Spes, 22 | Pastoral Constitution on the Church in the Modern World | Second Vatican Council | Promulgated by His Holiness Pope Paul VI, December 7, 1965.
Such is the mystery of man, and it is a great one, as seen by believers in the light of Christian revelation. Through Christ and in Christ, the riddles of sorrow and death grow meaningful. Apart from His Gospel, they overwhelm us. Christ has risen, destroying death by His death; He has lavished life upon us^{[cf. The Byzantine Easter Liturgy.]} so that, as sons in the Son, we can cry out in the Spirit; Abba, Father.^{[cf. Rom. 8:15 and Gal. 4:6; cf. also John 1:22 and John 3:1-2.]}

Please see Evil | New Advent the section The solution of the problem has been attempted by three different methods.