I was recently presented with a question: "If belief in the proposition 'Christ died for me' is not part of the content of the Gospel, how is it possible for a sinner to be relieved in his conscience concerning the forgiveness of his sins?"
In addition to my comments below, I also recommend reading my short article "Christ Died for Me" is NOT the Gospel where I basically address a similar issue.
Before answering the above question specifically, I think I should say something about the extent of the atonement. Nobody really believes in unlimited atonement to the extent that they'll say the sacrifice of Christ was both sufficient for all and efficacious for all. 99% of Christian Universalists don't even believe this. If there is even one minute of punishment for even one sinner in the afterlife, then there is no such thing as unlimited atonement in this strict sense. Since Scripture teaches that some people will be punished in the afterlife, the atonement must be limited, and this means that no one who has not yet believed the Gospel can assume that "Christ died for him" individually in a sufficient and efficacious manner.
I will now present how a sinner can be relieved in his conscience concerning the forgiveness of sins without making the proposition "Christ died for me" part of the Gospel:
If anyone believes that Jesus is the Christ, the Son of God, it means that he believes that He is the only Savior (John 4:42 KJV, John 8:24, Acts 4:10-12). This means that he believes that He is his Savior, if he is to be saved at all (Matt. 14:30). But this is precisely what it means to believe in Him as Christ (John 6:68-69). And by believing in Him as Christ, the believer can be assured that Christ did save him by His sufficient and efficacious work on the cross, for it is to such believers that justification, forgiveness of sins, and eternal life belong (Isa. 53:1ff, John 20:31, Acts 13:38-39).
From Matthew to Revelation, the Apostolic Gospel, if summed up in one brief proposition, was that Jesus is the Christ, the Son of God. Whoever believed this proposition was saved. Done. Finished. Full stop. We must not add anything to it (Deut. 12:32). The problem is that the vast majority of Christian churches regard belief in this proposition too small of a thing, revealing that they don't really believe it at all. Instead, the Christian churches insist that the merits of Jesus (or more precisely what they perceive to be the merits of Jesus) must be received or appropriated in order for them to attain their desired righteousness. And thus they appropriate things like "Christ died for me" as a qualifying condition through some mysterious, subjective, and elusive thing they call "faith," which they almost always ascribe to the Holy Spirit working in their heart so as to obfuscate the insidiousness of moving the focus from what Jesus did for sinners to what the sinner does to justify himself. In addition to this, they often add other conditions, too, such as water baptism, public confession of Jesus as Lord, and lifelong discipleship, saying that there is no justification where these things are absent, and thereby again revealing that they don't actually believe that Jesus' work on the cross was sufficient and efficacious enough to present even the most guilty of sinners acceptable in the sight of God.
I don't mix the Apostolic testimony with any qualifying act performed by me (or supposedly performed in me by the Holy Spirit), whether in thought, word, or deed. Christ is wholly Savior and I am wholly an object of His sheer mercy, having nothing to commend me to Him (Luke 18:9-14). Faith recognizes His righteousness and sees His glory, for these things are "revealed" by belief in the Gospel, apart from all those other things mentioned above (Isa. 43:10-12, Luke 7:29, Rom. 1:16-17, 3:21-26, 2 Cor. 4:3-6, Eph. 1:6). This Gospel, namely that Jesus is the Christ, the Son of God, is utterly at odds with the gospels preached by the mainstream Christian churches, whether they be Reformed, Arminian, Free Grace, or otherwise (Deut. 31:27-29, Acts 20:29-31, Gal. 3:1, 2 Tim. 1:15, 4:3-4). The Apostles never required anyone to appropriate to himself the status of the elect by convincing himself through some mystical heart work that "Christ died for him" individually in order to make true what wasn't true until it was appropriated. Instead, the proposition "Christ died for me" is something believers come to understand as an implication of believing the Apostolic Gospel, meaning that it was true for them even before they believed, since its validity does not rely on them performing some act, whether it be faith, appropriation, or whatever, but entirely on the sufficient and efficacious work of Christ for them at the cross (Gal. 2:20-21).