To attach forgiveness to atonement means you don't think that God is forgiving by nature. We are told that we are to forgive 70 x 7. Does God expect us to be more forgiving than He is. Do we have people saying that the mercies of mankind are eternal? Really?Good example. "Cleansing" is HOLINESS language, whereas "Forgiving" is RELATIONAL language.
I went through a long discussion about Atonement theory with a brother who came to Christ from a Buddhist perspective, which prompted me to do some research in the Theology books. One of the things about Penal Substitution theory that doesn't make sense is WHY Jesus's death would be necessary for forgiveness? If you're going to forgive somebody, forgive them. And before the usual howling about "God's Justice demands that somebody must pay!", that's easily refuted biblically by Ezekiel 18:20 ("The person who sins will die. The son will not bear the punishment for the father’s iniquity, nor will the father bear the punishment for the son’s iniquity; the righteousness of the righteous will be upon himself, and the wickedness of the wicked will be upon himself.") and Jesus's parable of the unforgiving servant who would have to be considered in the right for demanding payment of a 500 denarii debt.
Atonement itself, going back to the OT sacrificial model, IS about holiness. But yet I do see some overlap with forgiveness. I also see IDENTITY. Jesus in His death identified with us in our sinfulness. We in our lives are to identify with Him.
I think people underestimate how much God has to put with. Look at all the sin in the world. And it's still not yet enough to overturn God's mercy. But there is coming a day when God will say ENOUGH.