We're born again by regeneration, BECAUSE of the DEATH and Resurrection of Jesus Christ.
I'm not sure what your point is in trying to separate the Gospel unto Salvation.
You cannot have life until you acknowledge He died to pay YOUR sin debt. NO ONE can be saved without that fact.
1 Peter 2:24 New International Version (NIV)
24 “He himself bore our sins” in his body on the cross, so that we might die to sins and live for righteousness; “by his wounds you have been healed.”
Romans 1:16 New King James Version (NKJV)
16 For I am not ashamed of the gospel of Christ, for it is the power of God to salvation for everyone who believes, for the Jew first and also for the Greek.
Paul preached the WHOLE Gospel. He didn't go around saying " Believe in the Life of Jesus".
No. He preached Jesus's atoning DEATH and resurrection. He said he strove to know nothing but Christ, and Him crucified.
1 Corinthians 2:2 New King James Version (NKJV)
2 For I determined not to know anything among you except Jesus Christ and Him crucified.
1 Corinthians 1:18 New King James Version (NKJV)
Christ the Power and Wisdom of God
18 For the message of the cross is foolishness to those who are perishing, but to us who are being saved it is the power of God.
And just so we're clear what THE Gospel unto Salvation is by which we stand, that Paul CLEARLY states:
1 Corinthians 15 New King James Version (NKJV)
The Risen Christ, Faith’s Reality
15 Moreover, brethren, I declare to you the gospel which I preached to you, which also you received and in which you stand, 2 by which also you are saved, if you hold fast that word which I preached to you—unless you believed in vain.
3 For I delivered to you first of all that which I also received: that Christ died for our sins according to the Scriptures, 4 and that He was buried, and that He rose again the third day according to the Scriptures, 5 and that He was seen by [
a]Cephas, then by the twelve. 6 After that He was seen by over five hundred brethren at once, of whom the greater part remain to the present, but some have [
b]fallen asleep. 7 After that He was seen by James, then by all the apostles. 8 Then last of all He was seen by me also, as by one born out of due time.