By what mechanism does Faith lead to Salvation?

The short answer, I would say, is that we are saved by being part of what Christ did, and faith (which is, essentially, trust) is necessary to become part of what he did. Then, being baptised as a visible expression of our faith, we have died in company with Christ and are resurrected in company with Christ.

This is spelled out most clearly in Romans ch6. We have "died to sin" (v2). How? Because "all of us who are baptised into Christ were baptised into his death" (v3). We become part of him, we share in what he did. The result is that "If we have been united with him in a death like his, we shall certainly be united with him in a resurrection like his" (v5). "If we have died with Christ, we believe that we shall also live with him." (v6). "So you also must consider yourselves dead to sin and alive to God IN Christ Jesus" (v11).

I take that "IN" very seriously, because I think it is the key. We are saved IN Christ. This is not quite the conventional answer, because I talk about "inclusion" rather than "substitution".

In 1 Peter ch3 v20 and 2 Peter 2 v5 there is a comparison between being saved in the days of Christ and being saved in the days of Noah, part of the point of the comparison being that seven others were saved by accompanying Noah. In effect, he carried them through the waters. It is faith that puts us on the boat which Christ is steering through the flood,

Disclaimer: This Answer is my own interpretation that does not necessarily correspond to any specific denomination. That said, I believe it to be consistent with non-Calvinist Evangelical Protestantism.

C. S. Lewis once observed:

There are only two kinds of people in the end: those who say to God, "Thy will be done," and those to whom God says, in the end, "Thy will be done."

He is making a critically important point; God does not force people to believe, or to spend Eternity with Him. There most certainly are people that don't want to be with God.

As correctly noted in the question, Christ already died for all people. Scripture teaches that Salvation is a free gift, available to all. Therefore, we might imagine Salvation as being like a box overflowing with "eternal life: admit one" tickets that never empties. It's just sitting there, inviting people to take a ticket, with no strings attached.

In what manner, then, is faith required? Simply this: "faith" is nothing more than walking over and taking a ticket. Faith is believing that the ticket is real, and that there's a reason to take it. Faith is remembering, when you find yourself standing at the gate, that you have a ticket. Faith is choosing to walk through the gate rather than turning around and leaving. Someone without faith is someone that either ignores the tickets, or took one and later ripped it up.

Salvation without faith would be like God running you down and tackling you in order to staple a ticket to your forehead, then dragging you, kicking and screaming, to the gate and shoving you through. As noted earlier, God doesn't do that.

Faith isn't a chore one must complete, it's the state of accepting or rejecting the free gift of Salvation. That being the case, asking how it can be "required" doesn't entirely make sense. It's almost like asking "how can I enjoy the taste of chocolate when I refuse to taste anything?".

All branches of Protestantism, be they 'Evangelical' or not, ought to agree that faith in the resurrected Jesus Christ is essential for salvation. I have included the word 'resurrected' because it is faith that, not only did he die for sins, but was resurrected (due to having died without having sinned himself) that means he is still alive. Therefore, Christians do not have faith in a dead Jesus; it is the power of his resurrection that 'enables' salvation to come to repentant sinners.

You asked what 'the mechanism' is for faith leading to salvation, and fortunately for all who believe the Bible to be God's inspired word, the explanation is given there. Let this Reformed Protestant scholar explain in his book. First, he sets before us the reality of the horrific offense of the cross of Christ, for if this is not faced up to squarely at the outset, we will not awaken to why it was necessary, and what Christ's death on it accomplished for sinners:

"The impression is often given that the evangelical understanding of the cross, our doctrine of the atonement, somehow increases the pain of the Saviour. But it is not a theory that constitutes the pain of Christ. The pain was in the facts: that on the cross he suffered in body, suffered in soul, suffered from Heaven and from earth and from Hell. The fact is, Christ died. The fact is, He paid the wages of sin. The fact is, He was dealt with as sin deserved. There are three further facts that accentuate this central fact of the suffering of Christ.

First, His sinlessness...Secondly, the One who receives the wages of sin is the Son of God... The third twist in the paradox is that the cross is the act of God the Father: 'God so loved the world, that he gave his only begotten Son' (John 3:16). He did not spare His own Son but delivered Him up for us [believers] all (Romans 8:32)...

We really must learn to see the cross as a gigantic problem, a problem of mind-boggling proportions. We must see the scandal of it. We must say, 'What is this ugliness? What is this blot on the moral universe? What is this when God the Father is crucifying His own Son, when the righteous God is exacting from the Sinless one the wages of sin? What is this horrendous anomaly, this thing that we cannot understand, this offense, this hateful, wretched, ugly thing?'...

What is the New Testament solution to the problem? It is this: that the death of Christ was a sacrifice. It was not an accident. It was not an act of divine malice...

But what did it mean for Christ to be a sacrifice: to be the 'lamb without blemish and without spot' (1 Peter 1:19)? It meant that He was the One to whom sin was imputed... But sin was such, the offence against God such, the depravity such, the guilt such, that the salvation could not be completed until Christ had... endued what our sin deserved. The Son of man came 'to give his life a ransom for many' (Matthew 20:28; Mark 10:45). He was 'obedient unto death' (Philippians 2:8)... And what is death? It is the penalty for sin!" A Faith To Live By, Donald Macleod, pp. 131-135 Mentor, 1998

It is who we need to put faith in, and in what he accomplished for us, that matters, if we are to be saved. It is not just an abstract something - 'faith' - as if it is a quality that enables people to say, "Oh, yes, I have faith!" Much that passes for evangelical Christianity has tried to sanitize the cross of Christ, to gloss over it quickly in order to dwell on the niceness of the prospect of Heaven. But unless we know what was achieved on that cruel cross, and why that had to be done - for us sinners - we will have a shallow view of salvation due to having a shallow view of sin, and a superficial awareness of spiritual need. We need to begin to grasp the depth of our own depravity and the extent of our own guilt to truly appreciate God's provision in the blood of His Son.

'God has reconciled us to himself by Jesus Christ... For he has made him to be sin for us, who knew no sin, that we might be made the righteousness of God in him.' (2 Corinthians 5:18-21) And because he had no sin, death could not hold him. He arose in triumph over sin, death, and hell. That is the 'mechanism' of God's salvation in Christ. That is what, and who, we have to utterly trust - and to have faith in nothing, or no-one else for salvation. We cannot have faith in any religious systems, or in any creeds, or in any rituals, as if diligence in those respects will merit us God's salvation. We cannot contribute to the finished work of Christ on the cross. He did it all. If that doesn't get us down on our knees in repentant faith, nothing else will.

That is why it is vital to see that scripture states that Christ suffered for the sins of his people. There is no 'universal salvation' just requiring individuals to complete the work by their choice, their agreement. We must be clear why works cannot possibly save and why salvation is dependent upon faith in the finished work of Christ. This is the wonder of the cross.

Faith is a person's reaction to the death of Christ. It is the belief in/of something (Heb 11:1-3).

For example, at the first Passover in Egypt, they were to slay the lamb (Ex 12:6) and apply the blood (Ex 12:7). Slaying a lamb was necessary, but not absolute. One had to react in faith as faith to apply the blood, the sign (Ex 12:13).

Some believe in universalism, which is the idea that Christ died for all and all will be saved. But this theory leaves out the reaction, the faith to believe what is offered (salvation).

The Bible says Christ paid the price for many (Mt 26:28, Heb 9:15). Those are the ones who hear the gospel and react in faith, believing.