What is the basis for the argument that Jesus DID NOT have the ability to sin or give in to temptation?

The argument is that Jesus was fully God, and thereby His nature not only was in line with God's it is God's. Since God cannot sin and Jesus is God, Jesus could not sin.

The fact that He is fully God and fully Human enabled Him to experience our human weaknesses, while His holy, perfect, sinless nature enabled Him to experience them without succumbing to them.

A complete theological statement would be "because Jesus was fully Human, He had the capacity to sin, but because of His perfect nature, also being fully God and therefore perfectly good, He could not sin."

Both are true, but people have a hard time wrapping their head around that, choose half the truth, and fall into error.

Of the people who believe Jesus was capable of sinning, there are some who have the mistaken notion that in order to qualify as "real" and "genuine" a temptation must have at least a chance--regardless of how miniscule the chance might be--of causing the person who is being tempted, to sin. In other words, according to that notion, if there is no possibility whatsoever for a person to fail the temptation, then the temptation does not qualify as a temptation.

That notion is wrong not only as it applies to us mere mortals, each of whom has given in to temptation at least once in our lives, but also as it applies to the eternal Son of God who never once gave in to temptation.

The writer to the Hebrews makes this very clear:

For we have not a high priest who is unable to sympathize with our weaknesses, but one who in every respect has been tempted as we are, yet without sin (4:15 RSV, my italics).

For Jesus simply to sympathize with us in our weakness, all he would need to do is experience the same temptations we experience. There is no need for him to be able to (or have the capacity to) give in to the temptation for him to feel what the temptation is like, nor for that matter do we have to give in to the temptation in order to say we've been tempted. Just as Jesus chose not to give in, so we too can choose not to give in. Where we and Jesus differ in this regard is that we inevitably gave in to temptation and sinned, all of us just once, and I imagine most of us many, many more times than once.

Where Jesus was different from us in this regard is that he alone among all the human beings ever born was born of the seed of a woman and not the seed of a man. Jesus' conception in the womb of the virgin Mary was a singular, never-to-be-repeated event. All other human beings who have ever lived (or ever will live) were conceived when the seed of a man united with the egg of a woman. Jesus, however, as Genesis 3:15 put it in prophetic form, would one day be "the seed of the woman" and would deal a death blow to the original instigator of rebellion against God, none other than Satan himself.

As the first and representative man for the human race, Adam, when he sinned by disobeying God he not only signed his own death warrant, which God had promised would be the punishment for disobedience, he also bequeathed death to each subsequent generation. The Bible puts it this way:

. . . in Adam all die . . . (1 Corinthians 15:22a RSV).

Paul also laid out in lawyer-like fashion in Romans the following phrases (which I've taken the liberty to excerpt from Chapter 5):

• through one man sin entered into the world, and death through sin, and so death spread to all men, because all sinned— . . .

• by the transgression of the one the many died

• judgment arose from one transgression resulting in condemnation

• by the transgression of the one, death reigned through the one

• through one transgression there resulted condemnation to all men

• through the one man’s disobedience the many were made sinners

• sin reigned in death

Jesus, on the other hand, was not a child of Adam. Paul refers to Jesus as the second Adam who alone did what the first Adam could not do; namely, live a life of perfect obedience to God. Jesus' perfect obedience made possible his substitutionary death as "the Lamb of God who bears away the sins of the world" (John 1:29 and 36).

In short, as you observed in your question, "the redemption of mankind was never really in doubt," and the reason is that Jesus, who knew no sin became sin for us on the cross, so that in him we might become the righteousness of God (2 Corinthians 5:21).