Salvation to those who think they can lose it depends entirely on them.
If it did not depend on them, then they would not fear losing their salvation.
If any part of salvation depends on us, then all of it depends on us.
I don't think these people believe they needed the cross at all because ultimately they determine their salvation by their own works.
Ephesians 2:8-9 "8 For by grace you have been saved through faith; and this is not of yourselves, it is the gift of God; 9 not a result of works, so that no one may boast."
Did God lie when He said no one can save themselves, that there is no good in man that could save him.
Romans 7:18-19 "18 I know that nothing good lives in me, that is, in my flesh; for I have the desire to do what is good, but I cannot carry it out. 19For I do not do the good I want to do. Instead, I keep on doing the evil I do not want to do.…"
People say, "He is a good person, look at all the good he does." What they don't know is the heart of the one who "does the good." People who are not saved do good to glorify themselves, so they are not good; they only want to be seen as good by others.
The person who walks with Christ, doing authentically good things in obedience to Christ, is doing it through the Holy Spirit, and God gets the glory.
Romans 3:10-12 "as it is written: 'None is righteous, no, not one; no one understands; no one seeks for God. All have turned aside; together they have become worthless; no one does good, not even one.”
Since no one is good and cannot save themselves, then how could they do enough to gain heaven? Their good deeds are as filthy rags to God because only He is good and could do what was needed to save mankind, which was to shed His innocent blood on the cross to pay for our sins. We couldn't shed our own blood to save ourselves, so how could we possibly do anything short of shedding blood to earn or keep salvation?
Is a good deed equivalent to the shedding of blood? Of course it would be silly to say that. Since we cannot shed our own blood because we are "wicked without remedy", and Christ had to pay the price for us by shedding His blood, how could we think our silly little selfish good works save us on any level?
Those who think they are saving themselves by their good deeds or keeping their salvation by good deeds are trusting in themselves and not Christ. Ultimately, in their mind, they are in control of their own salvation, which is blasphemy.
Romans 3:10-11 "10 As it is written: “There is no one righteous, not even one. 11 There is no one who understands, no one who seeks God."
Galatians 3:10 "All who rely on works of the law are under a curse. For it is written: “Cursed is everyone who does not continue to do everything written in the Book of the Law.”