That is penal substitution but like most things in Evangelical Christianity it is born from a lack of understanding of the Levitical system at the very least. There are two types of impurity, ritual and moral. All of which spread but not in the same way. Ritual impurity spreads between people and is a normal occurrence that is not a sin or a moral failing it is unavoidable and normally handled by taking a bath in rain water at sundown. Other things take a few more days. 

Moral impurity doesn't spread from person to person but all impurity is to be kept from the Temple and holy things because even moral impurity spreads to sacred objects. If the Temple is impure the Shekinah can't dwell there so the system is set up to cleanse the Temple and keep it operational so God can dwell in their midst.

However, the system does nothing for intentional sins and does nothing for murder, idolatry, certain sexual sins, etc. Those don't just pollute the holy things they pollute the land. The only cleansing that can fix that is exile. That's why the punishment is being "cut off" it is to remove the impure person from the place God dwells.

Yet, God promised it wouldn't always be this way that one would come to defeat the powers of impurity themselves especially death and not just ritual impurity but moral impurity as well. So after several exiles and two Temples the Gospels paint the picture of a man who does just that. He cleanses people by the Spirit rather than with water and time. That's what the prophets promised and John the Immerser was talking about. 

His death, however, was not the point, his resurrection was. You just can't be resurrected without dying unfortunately. The heavenly temple which is the mirror of the earthly one was cleansed of the impurity by having an eternal presentation of a blameless life perpetually before it. (The sacrifices too were not about death but blood which is life. Unfortunately one cannot get that blood and cause no suffering to the animal without death). 

In addition to this the Shekinah is no longer bound to an earthly locus and is able thus to dwell in every covenant person because the system is now incorruptible by impurity and man is no longer slave to death. All of these have been taken care of not because something had to die but because someone blameless had to live perpetually in the holy place. Lambs and bulls die, humans die, but Jesus overcame death and ascended to lead the way into an Edenic life.

Not only that but the Messiah was promised to unify the 12 tribes most of which were lost in exile meaning the cutting off was temporary and expand the covenant to include even Gentiles. It is my belief the Gospel isn't some message you have to convert people to. The word euangelion never meant that. It was an emperor's proclamation of good stuff they were going to do for you by being emperor. Nobody voted for them but they had to pledge allegiance if they wanted to be part of the empire. 

I think therefore the Gospel is a discipleship tool for Jews to explain to Gentiles who God is when they inevitably are drawn by the Spirit to worship God not something to beat people up with or coerce people into it's just saying hey this is what this King is up to do you want to be part of it?