Faith precedes a heart change, this can be demonstrated. Suppose you are on a jury. The prosecution shows very convincing evidence that the defendant murdered a child. You cannot help but feel animosity and anger toward the defendant, though he claims he had nothing to do with the crime.
The next day the defense shows a video proving the defendant was in another part of town at the time of the crime, plus they present several witnesses who were with him at the time. You are now convinced beyond doubt the defendant was innocent, and his testimony true.
Would your heart attitude toward the defendant change? Yes, you would now view him as a good person who is the victim of mistaken identity. You would probably feel very guilty for your previous anger and animosity toward him.
This is how faith changes the heart. When a man believes God hates him, he cannot help but feel animosity toward him. But when a person hears the gospel that Jesus indeed loves him so much that he died for him, and that he will freely forgive all who come to him, his heart is immediately changed. He is no longer afraid of God, he is no longer angry toward God, or views God as an enemy who would crush him. His heart is changed forever.
Faith precedes the heart change. No one will love Jesus until he is first convinced Jesus loves him.
Your analogy is based competely upon a wrong foundation and therefore your conclusions are equally wrong. The problem is not that man believes that God hates him as you suggest. The problem is that he loves darkness and hates light and God is light. The problem is that his heart is intrinsically evil and is the source of sin (Mt. 15:18-21). The problem is that both David, Jesus and Paul declare that there is none instrinsically good but one and that is God (Mt. 19:15; Romans 3:9-18 with Psalm 14:2). The problem is that there is NO MIDDLE GROUND between a good tree and a evil true - you are one or the other and the bad tree "cannot" bring forth "good" fruit and "faith" is a good fruit. The problem is presented by Job 14:4 and the question can anything good come out of something evil? His answer is no. That is why the solution is a "new" heart (Ezek. 36:26) not a reconditioned old heart.
This is not a problem of power of choice as man has that power of choice. The problem is that the will of man is nothing more and nothing less than the faculty by which the heart expresses itself. That is precisely why the two Greek terms translated "will" in the New Testament are "boulomai" and "thelema." The former is the will directed by thinking whereas the latter is the will directed by feelings.
The only reason the will cannot choose to hate darkness (repentance) and love light (faith) is because the human heart (thinking and feeling) are intrinsically evil. So the will is not the issue. What is the issue is a bad, evil heart and God's solution is to "give a new heart" not overhaul the old heart (Ezek. 36:26).