Good news of the kingdom is specific to Matthew. There it is what Jesus preached (4:23; 9:35; cf. "the good news of God" in Mark 1:14) so it hardly includes, let alone focuses on, "salvation through Christ's death".
But the use of euaggelion ("good news, gospel") is predominantly Pauline and mostly applies to what Paul calls my gospel -- where, indeed, the death and resurrection of "Christ Jesus" is central.
There were several "gospels" or "good news" in "early Christianity," as Paul himself testifies (2 Corinthians 11:4; Galatians 1:6ff). And whatever gospel a historical Jesus may have preached would have been different from Paul's.