You said:
"And since we know that God has not declared all men without exception to be not guilty, free from guilt, and announced righteous, then its settled that Christ did not represent all men without exception in His Life and death of the cross. He simply did not die for all men without exception."
1 First of all, then, I urge that supplications, prayers, intercessions, and thanksgivings be made for all people, 2 for kings and all who are in high positions, that we may lead a peaceful and quiet life, godly and dignified in every way. 3 This is good, and it is pleasing in the sight of God our Savior, 4 who desires all people to be saved and to come to the knowledge of the truth. 5 For there is one God, and there is one mediator between God and men, the man Christ Jesus, 6 who gave himself as a ransom for all, which is the testimony given at the proper time. 7 For this I was appointed a preacher and an apostle (I am telling the truth, I am not lying), a teacher of the Gentiles in faith and truth.
— 1 Timothy 2:1–7 —
God’s intention is to provide a payment for all people which is only effective when the individual savingly believes.
Provisional Atonement: Christ died for the purpose of providing payment for the sin of all people making it possible for any and all to be saved. God loves all and wants all to be saved. In His love for all, he sent Christ to provide payment for the sin of all. Belief in Christ is necessary, however, to receive the benefits of Christ’s death and be saved. The gospel should be preached to all, and, upon hearing the gospel, any can come because Christ died for the sins of all people in the world.
Tim. 4:10 – God is the Savior of all men, especially of believers. So, there is a sense in which Christ is savior of unbelievers (i.e., he died for their sin, though they reject His payment on their behalf), yet a special sense in which he is savior of believers (by faith, they receive Christ’s payment for their own sin).
2 Peter 2:1 – refers clearly to unregenerate people as “denying the Master who bought them, bringing swift destruction upon themselves.
Or as I like to say, conditional atonement.