The "gospel of the kingdom" is the "good news of a new world order." The new order doesn't come about through political restructuring or violent revolution, but it consists of the redeemed community. The redeemed are those who've been delivered from enslavement to sin and death. As they go about doing good in the name of Christ, the kingdom advances in the world.
The miracles and ministry of Jesus were manifestations of the advancing kingdom of God through which the lost are found, the sick receive care, and the oppressed are granted justice. While this new order is not fully consummated until Christ's return, in the meantime his followers make whatever difference they can, bringing the rule of heaven to bear on earth.
The Romans defined the gospel of a new world order as the Empire under Augustus, whereas Christ-followers define it as the Kingdom of God under Jesus. While Augustus established a new order by force, Jesus, ironically, achieved victory in what appeared like defeat. He conquered by the cross. There he paid the penalty for sin, releasing those who follow him from its enslavement. Having been laid in the tomb and raised from the dead, Jesus conquered death, thereby delivering his followers from the fear and finality of this enemy of humanity.
So the message of Christ's death, burial, and resurrection is certainly core to the gospel, for it is through these events that Christ delivers us from our enemies by defeating them. While there is no gospel without the cross, the empty tomb, and the resurrection, this is not the extent of the gospel. If it were, then Jesus would not have been preaching the gospel prior to these events. But Scripture says that Jesus did preach the gospel---the gospel of the kingdom---a new world order.
So when Jesus preached the Sermon on the Mount, for example, was he preaching the gospel? Absolutely, for he was heralding a new world order, a new community under the reign of God. It's an order in which people keep their promises and love their enemies, resisting temptations to deceive or objectify others. The Sermon on the Mount holds up the model of life in the Kingdom of God and is therefore "gospel."
Since the message of the death, burial, and resurrection of Jesus is crucial to the conquering of sin and death, this message is foundational to the gospel (1 Cor. 15:1-8). Because of the cross and the resurrection people are redeemed from the old order of sin and death, delivered into the new order of liberty and life. The gospel is neglected if the cross and the resurrection are conspicuously absent from one's preaching.
Mere moralizing is not gospel preaching. Stating the negative consequences of sin is not enough. It's not enough to say, for example, that fornication should be avoided since it could ultimately lead to disease or the second death. The primary motive for purity is because we've been bought with the price of Christ's blood so we are now his possessions (1 Cor. 6:18-20).
Having been redeemed by Christ's blood, delivered by his work on the cross, we are incorporated into the new order of humanity. We are now charged with the task of being agents of redemption in this world, helping to deliver others from what binds them.