Unlimited atonement does not mean all are saved. It means all are offered salvation.
1 Timothy 2:4 [God our Savior] desires all people to be saved and to come to the knowledge of the truth.
2 Peter 3:9 The Lord ... is patient toward you, not wishing that any should perish, but that all should reach repentance.
As we live in time, it is difficult to understand God who it not constrained by time. Human reasoning fails.
I know that I am saved because I have heard God's word and believed.
How do you know if you are saved? With the theory of limited atonement you run into a problem as you can never be sure if Christ died for you. You can not share the gospel with assurance. You can only tell someone that Christ died for them if God chose them for salvation.
Reading the parables of Jesus, you can plainly see that limited atonement is foolishness. All were invited to the wedding. The seed was cast everywhere.
With the seed specifically, the good news would be a lie if Jesus didn't die for everyone.
John 1:29b Behold, the Lamb of God, who takes away the sin of the world!
The reasoning of limited atonement is foolish. From some website "If Jesus actually stood in my place and bore my sin on the cross as the Bible teaches, then I can never be punished for that sin." Wrong. Whoever believes in the Son has eternal life.
The website continues, "In order for Christ's atonement to truly be a substitutionary or vicarious atonement, then it must actually secure a real salvation for all for whom Christ died." Wrong again. Truly, truly, I say to you, whoever believes has eternal life.
And more, "If Christ acted as a real and true substitute for those for whom He died, then all for whom He died will be saved." Wrong, wrong, wrong. For with the heart one believes and is justified, and with the mouth one confesses and is saved.
And finally, "To say that Christ died a vicarious death in the place of all sinners but that not all sinners will be saved is a contradiction." Still wrong. Nowhere in the Bible does it say that if Jesus dies for you, then you are saved. You are letting your human reasoning get in the way. let the one who desires take the water of life without price
So, if the offer is made, and it is up to man to choose or not to choose, then there is no hope in salvation, as one could change their choice.
If my salvation is up to God, then I have assurance.
Besides, the ones who believe are those chosen by God.
Finally, your interpretation makes the atonement only effective for those who believe as well. Otherwise, all would be saved.