I wouldn't put it that way. The atonement is a crucial aspect of Jesus' intercession on behalf of believers. According to Paul's word on the subject, God was pleased to hear Jesus' petition on our behalf because of Jesus' righteousness and his obedience even unto death. The Atonement Jesus offered God was his own blood taken into the heavenly temple, which perfected the conscience of the believer.
Christ's reconciled the world to God by His death and resurrection. That has nothing to do specifically with believers. It had to do with the sin of the world. I John 2:2. His sacrifice atoned for the sin of the world. It makes it possible that Christ, now as High Priest can forgive sin when we confess our sin. There is no unilateral forgiveness. It is a mutual joining of Christ and a person who answers His call to come.
The Holy Spirit is in this world calling all men to repentence. This phrase of itself condemns the idea of limited atonement.
The concept that many Christians miss is the idea that Jesus himself will choose who will be granted eternal life.
Depends on whose eternal life one is referencing. By His resurrection by virtue of His Incarnation, He will raise all men to an eternal existence. another reason limited atonement is an impossibility.
Now, those that see and believe will be granted eternal life with Him if they remain faithful. I Pet 1:3-5
If you are referencing those who don't believe, they will be judged also and suffer the second death, which is spiritual separation from God for an eternity.
Unless Jesus knows me, he says, I will not be granted eternal life. (Matthew 7:23)
Those that believed and were faithful on all things He will grant eternal life.
The Father has given certain people to his son Jesus and all that the Father has given to him will be raised. (John 6, and John 10)
The Father gave all things to Christ. Col 1:20, Then corresponding texts is John 10:39 which states that Christ will raise all things, He will lose none.
Then vs 40 says, those that believe and see, He will also raise them up and give them everlasting life.
These three texts condemns outright the whole idea that there is such a thing as a limited atonement.
Another you should read is I Cor 15:12-22. That is universal redemption, reconciliation of all things
that Col 1: 20, II Cor 5:18-19 are speaking about.
You can also check Rom 5:12, vs 18 also spells out universal fall of man and universal redemption of man.