What is the biblical basis for the belief that Jesus' death made satisfaction for individual sins?

I don't know if this is a valid answer to your questions since you asked for affirmative explanations, however, I must answer in the negative. I believe the Bible only supports the class action lawsuit view as it clearly states that Christ died to sin once for all:

Hebrews 7:27 27 He has no need, like those high priests, to offer sacrifices daily, first for his own sins and then for those of the people, since he did this once for all when he offered up himself. Romans 6:10 10 For the death he died he died to sin, once for all, but the life he lives he lives to God.

Related to your questions, there is a common miss understanding that Christ paid the debt for each and every one of our sins, but, in effect, the scriptures reveal that Christ's death affected, instead, the cancellation of our debt.

Colossians 2:11-15 11 In him also you were circumcised with a circumcision made without hands, by putting off the body of the flesh, by the circumcision of Christ, 12 having been buried with him in baptism, in which you were also raised with him through faith in the powerful working of God, who raised him from the dead. 13 And you, who were dead in your trespasses and the uncircumcision of your flesh, God made alive together with him, having forgiven us all our trespasses, 14 by canceling the record of debt that stood against us with its legal demands. This he set aside, nailing it to the cross. 15 He disarmed the rulers and authorities and put them to open shame, by triumphing over them in him.

We individually access this cancellation, as Col 2:12-13 describe, when we are buried and raised with Christ in baptism through faith. Paul echoes this same idea in Romans 6:3-6:

3 Do you not know that all of us who have been baptized into Christ Jesus were baptized into his death? 4 We were buried therefore with him by baptism into death, in order that, just as Christ was raised from the dead by the glory of the Father, we too might walk in newness of life. 5 For if we have been united with him in a death like his, we shall certainly be united with him in a resurrection like his. 6 We know that our old self was crucified with him in order that the body of sin might be brought to nothing, so that we would no longer be enslaved to sin.

The idea of the body of sin being brought to nothing so that we would no longer be enslaved to sin in verse 6 is the same that Colossians 2:11 describes as the circumcision not made with hands that happens when we are baptized and involves the putting off the body of flesh.

This separation between us and our bodies of sin/flesh indicates the separation of ourselves from the corrupting curse within our flesh due to the sin of Adam so that while our bodies are still enslaved to sin by means of a curse and destined to die, we ourselves are free to be destined for greater things being united with the New Adam who is the Christ.

This is explained as a clever legal maneuver whereby dying with Christ we void any obligations we had under the old covenant of death. We are now free to be obliged under a new covenant of life to our Savior Jesus Christ.

Romans 7:1-4 1 Or do you not know, brothers—for I am speaking to those who know the law—that the law is binding on a person only as long as he lives? 2 For a married woman is bound by law to her husband while he lives, but if her husband dies she is released from the law of marriage. 3 Accordingly, she will be called an adulteress if she lives with another man while her husband is alive. But if her husband dies, she is free from that law, and if she marries another man she is not an adulteress. 4 Likewise, my brothers, you also have died to the law through the body of Christ, so that you may belong to another, to him who has been raised from the dead, in order that we may bear fruit for God.

So Christ engaged in a successful class-action lawsuit through his death on the cross. We become members of that class and beneficiaries of the settlement individually by being united with his death and resurrection through our baptism into a covenantal relationship with Christ.

Please allow in my answer and mention that your question is not entirely specific to one thing. Within all Christendom there are actually two kinds of redemption referred to:

1. Redemption from hell and sin-related suffering, conditional on accepting and following Christ. (1st John 3:15-24)

2. Redemption from death, conditional that we are a descendant of Adam (1st Corinthians 15:22)

In the case of the second, all are redeemed as a class of mankind, but in the first, we are redeemed conditional to our faith and relationship with Christ.

With that distinction, I would repeat the answer of Steve, that for sin, our sins have always been transferred symbolically to Christ on an individual basis as in the old law (Leviticus 4, Acts 8:22). The measure of Christ's atonement and suffering (remember to include the Garden of Gethsemane and the implied withdrawal of God, the Father's presence on the cross, Matthew 27:46) is without quantification or qualification by human reckoning, therefore it may not be possible to understand if the burden of each person's sins were added cumulatively upon him or if the full measure of suffering was simply enough for all by some super-earthly measure, until God reveals such things.

Without knowing the mechanics of Christ taking these things upon him, we may only surmise that our redemption is individual by the requirement that we individually agree to relinquish our sin or our bonds to sin unto him.

If the intent of your question is along the lines of whether Christ could or would endure all if just one person were to be saved, we have little precedence with which to declare that, but that sentiment is unquestionably clear (ex. Genesis 18:32) that the Lord values even the least number of us.

The debate between what is universal or collective and what is specific and personal burns hot in the Book of Job. Eliphaz, Bildad and Zophar erroneously apply universal religious principles to the specific case of Job. Job, on the other hand, appeals to God for specific relief and a specific list of the charges against him. Job's appeal is concerning how God treats the individual, namely himself.

One example is Zophar's appeal to half-measures:

5 Oh, how I wish that God would speak,
    that he would open his lips against you
6 and disclose to you the secrets of wisdom,
    for true wisdom has two sides.
    Know this: God has even forgotten some of your sin. (Job 11:5-6)

Job does not want God to forget some of his sin; he wants God to forgive all of his sin. Zophar in the rest of the chapter mocks the idea that Job can ever attain much wisdom in this matter. Job's later response is:

13 “If only you would hide me in the grave
    and conceal me till your anger has passed!
If only you would set me a time
    and then remember me!
14 If someone dies, will they live again?
    All the days of my hard service
    I will wait for my renewal to come.
15 You will call and I will answer you;
    you will long for the creature your hands have made.
16 Surely then you will count my steps
    but not keep track of my sin.
17 My offenses will be sealed up in a bag;
    you will cover over my sin. (Job 14:13-17)

Job is seeking a savior who can cover over all his sin. Nothing less will do. In the end, God commends Job and rebukes the friends. This vindication therefore acknowledges that God will provide a way for all a person's individual sins to be forgiven. In my book Job Rises: Thirteen Keys to a Resilient Life, I match eight aspects of Christ's life and ministry to Job's separate pleas for relief. In a subsequent book I found and described a ninth. Job was defining all that a savior would have to be and do in order to rescue him from his many problems, including sin. Those nine features of Christ's ministry are listed in Job in Chronological order according to the life of Christ, and even include the title for that savior: "son of man" (as worded in the ESV and Hebrew but not the NIV). If you put that together:

1. Individual not collective relief
2. To be performed by a savior
3. whose attributes match Jesus
4. and whose chronology matches Jesus
5. and which plan was endorsed by God

Therefore the Book of Job proves the point. As a type of Christ, Job is then called upon by God to pray for and forgive the friends and for them to offer sacrifices in the presence of Job that their sin of presumption may be forgiven.

See the parables of the Lost Coin and the Lost Sheep:

Or what woman having ten silver coins, if she loses one of them, does not light a lamp, sweep the house, and search carefully until she finds it? When she has found it, she calls together her friends and neighbors, saying, ‘Rejoice with me, for I have found the coin that I had lost.’ (Luke 15: 8-9)

Again,

Which one of you, having a hundred sheep and losing one of them, does not leave the ninety-nine in the wilderness and go after the one that is lost until he finds it? (Luke 15:4)

In these two parables, Jesus gives irrevocable testimony to His commitment to redeem each sinner by his or her name. Does one need more explanation ?

The biblical basis for believing that Christ died and arose to 'deal with, once and for all' every individual sin of all having biblical, saving faith in Christ includes some of the following biblical points. Not all, by any means, but enough I trust to satisfy you that a fair attempt has been made to clearly show a biblical basis. I propose to state basic principles - the rational that builds up to this view - with some sample texts given for each principle.

1. Nothing any sinner can do can even contribute towards forgiveness of their sins (a particularly Protestant teaching). Either what Jesus did was sufficient to deal with every sin committed by every sinner, or it was insufficient. If it depends on anything sinners must do to contribute towards forgiveness, or to complete a process, then Christ's sacrifice was incomplete and nobody could ever be sure they will be saved. If, however, the Bible assures believers that they are utterly saved purely by the grace of God, through faith alone, in Christ alone, then the salvation of each such sinner is 100% guaranteed - no 'forgotten' sins, 'or 'unconfessed sins', or too-hideous-to-forgive-sins could ever prevent God's total salvation from anyone he determined to save. Sample Bible texts, all bold mine:

"For Christ is the end of the law for righteousness to every one that believeth... For if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved... Whosoever believeth on him shall not be ashamed... the same Lord over all is rich unto all that call upon him. For whosoever shall call upon the name of the Lord shall be saved." Romans 10:4 & 9-13

"And by him [Jesus Christ whom God raised again, without corruption] all that believe are justified from all things, from which ye could not be justified by the law of Moses" - Acts 13:39

2. No mere human being, even one that managed never to sin up to the point of their death, could thereby take away the sin of more than one other person. ('An eye for an eye, a life for a life' principle).

"None can by any means redeem his brother, nor give to God a ransom for him (for the redemption of their soul is precious, and it ceaseth for ever) that he should still live for ever, and not see corruption... But God will redeem my soul from the power of the grave: for he shall receive me." Psalm 49:7-15

3. The N.T. specifically names some of the individuals who had all of their sins freely pardoned. Timothy is assured by the apostle Paul that unfeigned faith was first in his mother and grandmother, and was also in him - four individuals saved and called with a holy calling, before the world began! (2 Timothy 1:5-9, Romans 1:1)

"Christ Jesus came into the world to save sinners; of whom I am chief. Howbeit for this cause I obtained mercy, that in me first Jesus Christ might shew forth all longsuffering, for a pattern to them which should hereafter believe on him to life everlasting." 1 Tim. 1:15-16 And Ananias was told by the Lord that Paul was a chosen vessel before he became a Christian (Acts 9:15). And if God saved the chief of sinners, no sin is beyond God's ability to forgive.

"Elect according to the foreknowledge of God the Father... to an inheritance incorruptible, and undefiled, and that fadeth not away, reserved in heaven for you, who are kept by the power of God through faith unto salvation ready to be revealed in the last time... yet believing, ye rejoice with joy unspeakable and full of glory: receiving the end of your faith, the salvation of your souls" " 1 Peter 1:15

4. Sin is so hideous, all sin must be removed by God. We cannot remove our sins; only God can do that, and he does so, saving individuals "to the uttermost that come unto God by [Christ]." - Hebrews 7:25.

"And one seraph cried unto another and said, Holy, holy, holy, is the Lord of hosts; ... Then I said, Woe is me! For I am undone; because I am a man of unclean lips, and I dwell in the midst of a people of unclean lips, for mine eyes have seen the King, the Lord of hosts. Then flew one of the seraphim unto me, having a live coal in his hand, which he had taken with the tongs from off the altar. And he laid it upon my mouth, and said, Lo, this hath touched thy lips; and thine iniquity is taken away, and thy sin purged." Isaiah 6:1-7

"But thou hast in love to my soul delivered it from the pit of corruption: for thou has cast all my sins behind thy back." Isaiah 38:17

"Thou art of purer eyes than to behold evil, and canst not look on iniquity" Habakkuk 1:13.

"Whosoever committeth sin transgresseth the law; for sin is the transgression of the law. And ye know that he was manifested to take away our sins; and in him is no sin... He that comitteth sin is of the devil... Whoever is born of God doth not commit sin... because he is born of God." 1 John 3:4-5

Summary of the case presented: God lumps together as a group all sinners. Not one of them can save themselves from their sins: "Both Jews and Gentiles are all under sin: as it is written, There is none righteous, no, not one." (Romans 3:9-10 quoting Micah 7:9.) "But the scripture hath concluded all under sin, that the promise by faith of Jesus Christ might be given to them that believe." (Galatians 3:22)

The only one who could save untold billions of sinners had to be more than 'just' a sinless human - he had to be both divine and human to be the one Mediator between God and men, the man Jesus Christ - 1 Tim. 2:5

But those whom God saves were ordained unto salvation even before the world began, and not just a group, but as individuals to be called by God in due time. Even those of faith in Old Testament times are named as individuals in Hebrews chapter 11, alongside those individual believers in the New Testament.

All of those individual's sins have to be removed, because God is too holy to look upon sin & Psalm 67:18 & Deut. 24:16. Nothing unholy or unrighteous can be in God's holy, righteous presence, so all those who will be before the throne of God must be utterly cleansed from all sin. That will be when God has created "a new heaven and a new earth, in which righteousness shall dwell" (2 Peter 3:13). No sin, which is unrighteousness, can dwell in God's holy, righteous heaven, for he IS righteous. Only what the Son of God did can achieve such a magnitude of cleansing.

Off the top of my mind, there are two passages upon which the belief of individual satisfaction can be based. One is from Paul's letter to the Romans, v 6:10 [RSV]:

While we were still weak, at the right time Christ died for the ungodly. Why, one will hardly die for a righteous man—though perhaps for a good man one will dare even to die. But God shows his love for us in that while we were yet sinners Christ died for us. Since, therefore, we are now justified by his blood, much more shall we be saved by him from the wrath of God. For if while we were enemies we were reconciled to God by the death of his Son, much more, now that we are reconciled, shall we be saved by his life.

While Paul writes partly in the collective, the use of the singular in the second sentence leads me to think that he is writing singly to an audience which is collective.

A second passage is from John's first epistle, v. 8-10. Again, written to a plural audience, but in my view with a mind towards addressing the individual members of that audience:

If we say we have no sin, we deceive ourselves, and the truth is not in us. If we confess our sins, he is faithful and just, and will forgive our sins and cleanse us from all unrighteousness. If we say we have not sinned, we make him a liar, and his word is not in us. [RSV]

In my mind, John 1 Chapter 2 is likewise written to a collective as the audience of the epistle, but is intended to be interpreted by Paul as applying to the individuals who constitute the collective.

Part of the solution here might be to look at the Old Testament sacrifices which Jesus is fulfilling. When an Israelite took his lamb to the temple to be sacrificed, he had no sense that his sacrifice was for all the sins of God's people collectively. The sacrifice was being made for his own sin. Take a look at Leviticus 4, for example.

Hebrews specifically connects these Old Testament sacrifices to the work of Christ. Just as the OT lamb was slain because of the specific sins of a specific individual, so too did Jesus die for the specific sins of his people.