I think that we need to separate two issues here. I think that we need to make a distinction between "belief in the gospel" and "believing that Calvinism is true." The second involves mental assent to the basics of Calvinism. The first involves more than mental assent (since we all hold as true the statement in James, even the devils believe). Belief in the gospel is more than just believing the aspects of the gospel to be true. It goes considerably further. I hold to the noetic effects of sin, so I hold that people cannot believe in Jesus or the gospel when their heart is in love with sin (see Jn 3:19). Sin's perversion upon the mind renders the natural man unable to properly perceive their lostness and consequently the amazingness of the cross.
However, depravity does not rule choices; rather, depravity demonstrates that choices aren't all that simple and depravity binds a person's will. However, when God makes His light to shine in our hearts, and then we actually perceive God's glory in the face of Jesus Christ (2 Cor 4:6), then the person will want to believe and thusly choose to believe.
We have Romans 4, and Paul's inspired presentation of Abraham and his faith. In 4:3 we see the quote from Gen 15:6. Abraham believed God, and it was credited to him as righteousness. Let's work from the obvious to the less obvious. In response to God's reiteration of the promise--that Abram's seed would be as numerous as the stars of heaven--Abram believed God. At minimum, Abram (in spite of the later evidence of his body and Sarah's womb being dead (Rom 4:19)) believed God. In Genesis 22, Abraham's faith took the form of action as he trusted God's promise in spite of God's own command seemingly coming into conflict with the promise, but he trusted that God could raise Isaac from the dead (Heb 11:17-19). Abram's faith took the form of action in Heb 11:8.
The opening post addressed the issue of evidence, so I'll mention a few things about that topic. Most notably we have Pharaoh (Exodus) and the truth suppressors in Romans 1. Pharaoh had plague after plague; he had evidence after evidence. But he did not believe. The truth suppressors have the testimony coming from the things that are made, so as to be without excuse, but the truth-suppressors push away the truth, proclaim to be wise in their own eyes (epistemological autonomy), and thusly they become fools (more demonstration of the noetic effects of sin). And Romans 1:18-32 is all about the wrath of God being revealed, until Romans 3:21 speaks of the righteousness of God being revealed. Again, an essential part of the gospel Paul is elucidating in 1:18-3:20 is that people are sinful and under the wrath of God, so that every mouth would be stopped and the whole world held accountable to God (3:19-20). An essential part of the gospel is man's sinfulness, accountability, and deserved wrath from God. The problem must be understood before the solution makes sense. But I must get back to the issue of evidence. Pharaoh and the truth-suppressors had it in abundance and still rejected it, so evidence alone is not saving. However, the proper perception of God's glory in the face of Jesus Christ in the gospel is something that God makes to shine in the heart of a person; and thusly God overcomes man's depravity by the same creative power He displayed when He said, "Let there be light."
Thoughts?