For RC and EO - What did Jesus' death on the cross actually achieve for you?
The council of Trent session six gave this information for those curious to know what the Catholic Church teaches about justification. It has relevance for your question.
CHAPTER I
THE IMPOTENCY OF NATURE AND OF THE LAW TO JUSTIFY MAN
The holy council declares first, that for a correct and clear understanding of the doctrine of justification, it is necessary that each one recognize and confess that since all men had lost innocence in the prevarication of Adam,[3] having become unclean,[4] and, as the Apostle says, by nature children of wrath,[5] as has been set forth in the decree on original sin,[6] they were so far the servants of sin[7] and under the power of the devil and of death, that not only the Gentiles by the force of nature, but not even the Jews by the very letter of the law of Moses, were able to be liberated or to rise therefrom, though free will, weakened as it was in its powers and downward bent,[8] was by no means extinguished in them.
CHAPTER II
THE DISPENSATION AND MYSTERY OF THE ADVENT OF CHRIST
Whence it came to pass that the heavenly Father, the Father of mercies and the God of all comfort,[9] when the blessed fullness of time was come,[10] sent to men Jesus Christ, His own Son, who had both before the law and during the time of the law been announced and promised to many of the holy fathers,[11] that he might redeem the Jews who were under the law,[12] and that the Gentiles who followed not after justice[13] might attain to justice, and that all men might receive the adoption of sons.
Him has God proposed as a propitiator through faith in his blood[14] for our sins, and not for our sins only, but also for those of the whole world.[15]
CHAPTER III
WHO ARE JUSTIFIED THROUGH CHRIST
But though He died for all,[16] yet all do not receive the benefit of His death, but those only to whom the merit of His passion is communicated; because as truly as men would not be born unjust, if they were not born through propagation of the seed of Adam, since by that propagation they contract through him, when they are conceived, injustice as their own, so if they were not born again in Christ, they would never be justified, since in that new birth there is bestowed upon them, through the merit of His passion, the grace by which they are made just.
For this benefit the Apostle exhorts us always to give thanks to the Father, who hath made us worthy to be partakers of the lot of the saints in light, and hath delivered us from the power of darkness, and hath translated us into the kingdom of the Son of his love, in whom we have redemption and remission of sins.[17]
CHAPTER IV
A BRIEF DESCRIPTION OF THE JUSTIFICATION OF THE SINNER AND ITS MODE IN THE STATE OF GRACE
In which words is given a brief description of the justification of the sinner, as being a translation from that state in which man is born a child of the first Adam, to the state of grace and of the adoption of the sons of God through the second Adam, Jesus Christ, our Savior.
This translation however cannot, since promulgation of the Gospel, be effected except through the laver of regeneration or its desire, as it is written:
Unless a man be born again of water and the Holy Ghost, he cannot enter into the kingdom of God.[18]