T'here's no evidence in any of the recorded messages from any of the apostles that they ever said the Christ died for everyone's sins while they evangelized. If it carried the importance in the way the OP say it does, it would be all through their messages. It would at least be mentioned once, you'd think.'
Yes, there is evidence!
1 John 2:2 tells us that He died for the sins of the whole world:
"And he is the propitiation for our
sins:
and not for ours only,
but also for the sins of the whole world."
Colossians 1:20 tells us that He made peace through His blood - to reconcile ALL things unto Himself:
"And, having made peace through the blood of his cross, by him
to reconcile all things unto himself; by him, I say, whether they be things in earth, or things in heaven."
Paul in
Romans 5:18 says that the free gift came upon
ALL men unto justification of life:
"Therefore as by the offence of one judgment came upon
all men to condemnation; even so by the righteousness of one
the free gift came upon all men unto justification of life."
All men can be saved if they believe, because "by the righteousness of one
the free gift came upon all men unto justification of life."