My concerns are that "purification" is describing moral impurity and committed sin as well as our very "sinfullness" for lack of a better word.
The issue is our sinfulness (wickedness) as sins (actions) themselves are but manifestations or "fruits" of our wickedness. If that is what you mean, I agree.
Sins are forgiven, wickedness (unrighteousness) is cleansed.
I guess we can think of it like a crime. A murderer murders because he is a murderer (hate in his heart). Christ identified the issue in Matthew 5.
There is absolutely no objection that I have found (and I ask again for anyone who knows otherwise to show me) of any Calvinist or other advocate of PSA to the idea of us being purified or washed and that it can be described as done by the blood of Christ, often without mention of his death or his specifically "taking" our sin onto himself.
I doubt you will find any objection from PSA theorists as we all know that Christ's blood cleanses from all unrighteousness.
The problem comes in when people define the atonement according to what is NOT in the Bible but they believe is taught by the Bible.
In the Bible sins are forgiven (think of the scapegoat) and wickedness is cleansed. The idea that sins are transferred to Jesus and then punished so that we can escape the wrath to come is not logical or biblical.
Sins are actions. They are things that have been done. Christ bears our sins (ultimately bears the consequences of our sins). But sins are themselves superficial (sins are the bad fruit produced by the bad tree). The fruit of sin itself is death. "Sin produces death", "the wages of sin is death". We will all suffer this death for all have sinned. It is appointed man once to die (because of sin) and then face God's judgment (the "wicked are storing up wrath for themselves for the day of wrath"; "the wicked will not enter the kingdom of God"; "God's wrath is upon the wicked").
In the Bible "atonement" is not substitution but "cleansing". The high priest made atonement for the tent of meeting, then the altar, then the sins of the people. The Day of Atonement was to be observed as a time of cleansing (Leviticus 16).
The priest took the blood (life) from the animal into the Tabernacle. He made atonement for the tent of meeting, the altar, and finally the sins of the people with this blood (life).
You are right that there are two distinct aspects the Atonement deals with. They are physical death as the death sin produces as a wage and then the judgment of God against the wicked "on that day".
But you are blending the two together when it fits your theory and separating them when it doesn't. Scripture does not do this.
Our sins are taken away (removed). It is as if we did not do them. God remembers them no more. Yet we still suffer the death those sins produced (we still suffer under the power of evil, the dominion of Satan). Jesus shared in our humanity, became flesh, even though He knew flesh will not inherit the kingdom of God.
What you are missing is the "why", or the basis for our forgiveness. It is not that God all of a sudden decided to compromise His justice snd punish the innocent or clear the guilty. Had God done that He would cease being God as He has revealed Himself in His Word.
The reason we can be forgiven is not because God punished the Righteous but because Christ Himself is the Guaranteor or Surety of our faith. We will be recreated, conformed to His image. Jesus IS the reconciliation of God and man.
This goes to God's judgment rather than the death all are appointed to die because of their sins. He reconciles us. He is the Atonement that cleanses us from all unrighteousness.
That's actually the work of the Holy Spirit, described in scripture and explained in great detail by Owen. It's precisely why I keep saying that this was a unique redemption plan, involving the whole Trinity, that was devised before the foundation of the world. The verse you keep using serves to show just how special the circumstances and those involved (the Triune God) were in making this possible, the just for the unjust, how that Christ died for our sins according to the scriptures. (Christ died for us according to the scriptures as a purification ritual would be insufficient as an explanation).
It was a unique redemption plan. I do not know why you keep going back to this being God's righteousness manifested through the law. It was something else, something promised but obscured in the Old Testament.
It is the work of God (God takes out our old heart, gives us a new heart) but this is uniquely "in Christ" (we are made new creations in Christ). The old goes away and the new comes.
I believe that Christ died for our sins according to the Scriptures (rather than an idea NOT in the biblical text, not "the faith once delivered).
This matches the Levitical system because that system foreshadowed or was a witness to the salvation to come.
And in the Levitical system there was no substitution. A sacrifice was killed, this provided the means by which the priest made atonement for the tent of meeting, the altar, and the people's sins by cleansing or purification.
Jesus likewise entered the Most Holy Place not made with hands through His own blood as a Propitiation to be recieved by faith (His blood cleanses from all unrighteousness).
The second aspect of the Levitical system was a goat that was not sacrificed (a live goat). This is the animal set aside to Azazel (the "scapegoat", "totegoat"). This goat was sent to take away the sins of the people, to symbolically remove the consequences of sin.
Christ suffered these consequences with us, is the Lamb who takes away the dins of the world. But rather than by nullifying Scriptute it is by giving Life (the sting of deatg is removed). The Second Adam became a Life giving Spirit.
The biblical text itself makes sense without wondering "but, what does it really teach". The Bible teaches "what is written", it teaches the biblical text. Anything ekse is nothing but human wisdom (philosophy).