Can it be shown in the New Testament when a person believed in Jesus Christ, but their faith was rejected because not deep or sincere enough?
We must believe in Jesus Christ as the Son of God, not just agree that He exists.
Automatic salvation indicates God performing what He has promised.
Aside from individuals pretending to believe, or believing superficially, and not truly exercising genuine faith, I see no reason to think that once we do believe in Jesus Christ, more has to happen to achieve salvation. Authentic faith must produce fruit or works that coincide with faith, but we are saved by grace through faith.
Nowhere in scriptures do I see anything like “believe in Jesus Christ as Son of God and your personal Lord, then maybe God will credit that faith as righteousness, but maybe He will not.”
Acts 8:12, 37
12 But when they believed Philip as he preached the things concerning the kingdom of God and the name of Jesus Christ, both men and women were baptized.
37 Then Philip said, “If you believe with all your heart, you may.”
And he answered and said, “I believe that Jesus Christ is the Son of God.”
Romans 4
3 For what does the Scripture say? “Abraham believed God, and it was accounted to him for righteousness.”
5 But to him who does not work but believes on Him who justifies the ungodly, his faith is accounted for righteousness,
9 Does this blessedness then come upon the circumcised only, or upon the uncircumcised also? For we say that faith was accounted to Abraham for righteousness.
11 And he received the sign of circumcision, a seal of the righteousness of the faith which he had while still uncircumcised, that he might be the father of all those who believe, though they are uncircumcised, that righteousness might be imputed to them also,
13 For the promise that he would be the heir of the world was not to Abraham or to his seed through the law, but through the righteousness of faith.
16 Therefore it is of faith that it might be according to grace, so that the promise might be [a]sure to all the seed, not only to those who are of the law, but also to those who are of the faith of Abraham, who is the father of us all
23 Now it was not written for his sake alone that it was imputed to him,
24 but also for us. It shall be imputed to us who believe in Him who raised up Jesus our Lord from the dead,