Agreed. And the "Christian" who is doing "good works" merely to save his own eternal skin, is someone trying to save their own life, of whom Jesus said, " He who would save his life shall lose it, but he who loses his life for my sake and the gospel shall be saved."
But the "Christian" who would not lose his life (i.e. deny himself) for Jesus' sake and the gospel because He has been convinced that trusting in the substitutionary death of Christ and adopting some of the stereotypical traits of religious christianity ensures immortality, is also someone who is saving their own life, and he will not saved it. He will lose it.
We must believe in both the death of Jesus for our sins and His resurrection for our justification, by which resurrection He was authorised to give us the promised Holy Spirit, who empowers us to will and do what pleases God. By Him we are able to put to death the deeds of the flesh, that the life of Jesus be manifest in us.
We cannot divorce doing righteousness from being saved, as if being saved from our sins by faith in Christ's death for us releases us from any responsibility to keep on being saved from temptations and sinning, doing good by faith in His Life in us.
It is possible to believe I could potentially lose my salvation by rejecting the Holy Spirit's power in my life, without the potential to lose salvation in this way driving me to do good works out of an attempt to save my own eternal skin. I can, instead, yield to the Holy Spirit because I want to be as much as possible like the Saviour who loved me and gave Himself for me, and the Holy Spirit has been placed in me to disciple me into conformity to Jesus, by empowering me both to will and do what pleases God.