According to believers in the inexorable damnation of the unreached, why should they be punished in Hell and not be given more merciful alternatives?

My short answer to this is, because punishment in hell is what they deserve, and is in accordance with the nature of God (with the disclaimer that I believe the punishment is death/destruction, I do not believe eternal torment is deserved by even the worst of sinners).

The question seems to be a variation of another common question, which is 'How could a loving and good God send people to hell who lived good lives other than their unbelief', and the answer is similar to what Mike has said; we simply fail to appreciate the weight and severity of even the smallest sin. Your question is based on the lack of control of the unreached person in their unsaveability, but that person is still a sinner by nature and that sin has consequences, regardless of the availability or lack thereof of the Gospel where they are.

To give a more in-depth answer about the nature of God and why He must punish in hell instead of, as you say, giving everyone a second chance, or to take it even further, universalism, would require more knowledge than I possess, but I believe work has been done on this subject by others (I can't think of any works off the top of my head)

The best answer is one you may find the least helpful. If you read the book of Job, you see that Job goes through suffering that is inflicted by the devil, but permitted by God. Job cries out for answers, but at the end the only answer that God gives is that Job simply cannot understand God's actions or purposes. Nevertheless, the end result is that Job is blessed because of his faithfulness and uprightness, refusing to curse God even in the midst of His troubles, and only going to God for answers.

We see this also in the famous quote from Isa 55:8-9:

‘For my thoughts are not your thoughts, neither are your ways my ways,’ declares the Lord.
‘As the heavens are higher than the earth, so are my ways higher than your ways and my thoughts than your thoughts.

Jeremiah 18:1-6:

This is the word that came to Jeremiah from the Lord: “Go down to the potter’s house, and there I will give you my message.” So I went down to the potter’s house, and I saw him working at the wheel. But the pot he was shaping from the clay was marred in his hands; so the potter formed it into another pot, shaping it as seemed best to him.

Then the word of the Lord came to me. He said, “Can I not do with you, Israel, as this potter does?” declares the Lord. “Like clay in the hand of the potter, so are you in my hand, Israel".

Isa 29:15-16:

Woe to those who go to great depths to hide their plans from the Lord,
who do their work in darkness and think, “Who sees us? Who will know?”
You turn things upside down, as if the potter were thought to be like the clay!
Shall what is formed say to the one who formed it, “You did not make me”?
Can the pot say to the potter, “You know nothing”?

Similarly from 1 Cor 2:16:

‘Who has known the mind of the Lord so as to instruct him?’

But there Paul's point is actually a little different, which is that the servant of God who is filled with God's Holy Spirit is able to know the mind of the Lord, because he / she has the mind of Christ, which is what comes from trusting (pistis) in Jesus, which also includes acting on that faith / trust and obeying - remember the story of the wise and foolish builders (Mt 7:24). We know that judgement has been given to Jesus and that he will judge justly.

In summary, we don't fully know what happens to those who oppose God, but Jesus had strong warnings for those who reject Him, and we learn more from Him about hell and final judgement than anyone else. So the question is, why not submit to your Creator, why not seek Him out and obey Him? That is the only way to understand (as laid out in scripture).

Once a person enters eternity, the imperfect passes away and the truth is known.

For now we see only a reflection as in a mirror; then we shall see face to face. Now I know in part; then I shall know fully, even as I am fully known.

13 And now these three remain: faith, hope and love. But the greatest of these is love. (1 Corinthians 13:12-13)

This introduces a problem. The righteous shall live by faith. Faith requires belief in what is not fully manifest - that which cannot be seen or fully proven. Once the truth is fully known, all possibility for the development of faith is lost. If you have it already, it flourishes because faith will remain. But if you do not have it, the conditions for developing it are gone. All who are without faith when they enter the afterlife are sealed into that condition.

Some religious philosophies other than Christianity have an answer to this problem. In Chinese folk religion, the tea of forgetfulness erases your memory. Then you are sent back for another life, called reincarnation.

A reincarnation-based system offers the possibility of entering eternity, learning the truth about your sinfulness and the ultimate reality, yet being put back into a situation where you get another chance to develop faith and pursue truth.

That is not the system that Christianity teaches. That system does not require God to die for us. We are given infinite time to get it right.

Instead of millions of small chances to advance, the God of Christianity made it possible to maximize just one chance.

Those other philosophies think the infinite gap between God's holiness can be bridged through many small advances. But only an infinite series of finite steps can reach infinity. That would mean we never actually get to perfection. We never actually get to Heaven.

In the Christian way, the true depths of sin are taken into account. Only an infinite sacrifice can bring us in a finite time to perfection and permit us to enter heaven.

The question is similar to Abrahams appeal to the lost in Sodom. Abraham does not want God to destroy the whole city but pleads for mercy:

Then Abraham approached him and said: “Will you sweep away the righteous with the wicked? What if there are fifty righteous people in the city? Will you really sweep it away and not spare the place for the sake of the fifty righteous people in it? Far be it from you to do such a thing —to kill the righteous with the wicked, treating the righteous and the wicked alike. Far be it from you! Will not the Judge of all the earth do right?” Gen 18:22-25

‘Far be it’ for God to kill innocent people, Abraham pleads. What about the babies laying in their cribs in that sizable city? They have never heard the gospel? How could God destroy them by fire? Surely God would not do this. So Abraham keeps pleading to lower the number of righteous souls in Sodom to avoid the terrible judgment:

Then he said, “May the Lord not be angry, but let me speak just once more. What if only ten can be found there?” He answered, “For the sake of ten, I will not destroy it.” Gen 18:32

Certainly there were more than 10 infants in the city so infants are not innocent. The same goes for the infants not saved in Noahs Ark. The problem is that all born of Adam are cursed and deserve death, there are no innocent souls. If the whole of all humanity were punished eternally it would be just. So when we are aghast at the thought of seemingly innocent peoples unfairly treated (as though guilty souls deserve some certain portion of ‘mercy’ in a delay of their judgment, or even deserving ‘grace’ in their salvation) we show ourselves only that we do not understand the severity of sin.

The beauty of grace is it saves some among those who deserve eternal damnation:

For those God foreknew he also predestined to be conformed to the image of his Son, that he might be the firstborn among many brothers and sisters. And those he predestined, he also called; those he called, he also justified; those he justified, he also glorified. Rom 8:29-39

Having accepted these scriptures without wrestling their meaning away to conform to our own limited sense of justice, we are also bound to know that God loves all souls and does not desire to destroy those he destroys. He could not command us to love our enemies, as we follow his example, if he did not. How all these truths are resolved into the eternal councils of God no man can approach without entering into some sort of foolish speculation. Only faith obtains the gift and produces love and joyful hope.