Faith in Christ changes our relationship with God.
Yes it does. It aids our relationship with God. But God is not alienating himself from anybody because of sin. He alienates himself from those who choose to do wicked things, and if they "get born again" through faith in Christ, and don't show forth the fruits of repentance, that is, they continue in their wicked ways, God hides himself from them, even if they have all kinds of "faith in Christ."
Sinners are not alienated from God, and therefore don't need to be reconciled to God. God never left them. God does not condemn them. God just wants to set them free from the bondage of sin by eliminating the sin. THAT'S the salvation Jesus brought, first to the House of Israel, then to the nations.
Faith in Christ is a wonderful gift and certainly part of God's plan to make man in His image, but it isn't the only way to be in right standing with God. You can do what it says in the Sermon on the Mount without ever hearing of Christ or by following the teachings of Mohammed that say the same thing. "Blessed are the peacemakers" works for Christians, Jews, Muslims, and atheists alike.