I agree with that assessment. I don't think we get a clear picture in the NT about how atonement "works."
That said, I actually do think we are told about the mechanism of salvation in the NT, Paul especially.
How are we saved? We're saved by the Spirit. The Holy Spirit is the means and mechanism of salvation.
Rather than tour through all of Paul's letters, let's focus in on Romans 8 to see this illustrated.
According to Paul, the human predicament is that we are all slaves to Sin, death and the devil. We are dead, incapacitated and weak. Cut off from God's power, separated from God's life.
We are saved, liberated and rescued from our bondage by the power of the Holy Spirit. We are reconnected to God's life through the Holy Spirit.
Again, how are we saved? Answer: By receiving Holy Spirit. The Holy Spirit is the mechanism, the means of salvation.
Here's how Paul describes it at the start of Romans 8:
Romans 8.1-6The Spirit sets us frees from bondage ("Where the Spirit of the Lord is, there is freedom." 2 Cor. 3.17). The Spirit gives us the capacity to please God by walking in righteousness. The Spirit gives power and life to our mortal bodies.
There is therefore now no condemnation for those who are in Christ Jesus. For the law of the Spirit of life has set you free in Christ Jesus from the law of sin and death.
For God has done what the law, weakened by the flesh, could not do. By sending his own Son in the likeness of sinful flesh and for sin, he condemned sin in the flesh, in order that the righteous requirement of the law might be fulfilled in us, who walk not according to the flesh but according to the Spirit.
For those who live according to the flesh set their minds on the things of the flesh, but those who live according to the Spirit set their minds on the things of the Spirit. For to set the mind on the flesh is death, but to set the mind on the Spirit is life and peace.
For the mind that is set on the flesh is hostile to God, for it does not submit to God's law; indeed, it cannot. Those who are in the flesh cannot please God.
You, however, are not in the flesh but in the Spirit, if in fact the Spirit of God dwells in you.
Anyone who does not have the Spirit of Christ does not belong to him.
But if Christ is in you, although the body is dead because of sin, the Spirit is life because of righteousness. If the Spirit of him who raised Jesus from the dead dwells in you, he who raised Christ Jesus from the dead will also give life to your mortal bodies through his Spirit who dwells in you.
In short, if we were to ask Paul our questions--How are we set free? How are we given new life? How are we made into a new creation? How are we given the ability to walk in righteousness? How? How? How?--Paul's answer would be simple.
The Holy Spirit.