Evangelicals, I think, don't need to prove themselves loyal to God. You might have a more Catholic theology there. Where Catholics and Orthodox pray about "being made worthy to be heard in the presence of your divine majesty," and such, evangelicals would never choose to pray like that. The Reformation idea of "sola fides, sola gratia, sola scriptura" ("by faith alone, by grace alone, by Scripture alone") is very strong. It's not always thought through among us, logic not always being one of our strong points, but it is a stand that we take seriously, and thus we would never, ever say that we must prove ourselves to God. "Jesus did it all" would be more of our thinking.
Now, you are right about "original sin." We generally feel (literally--we don't always think about this very well) that we inherit a standing before God by virtue of our birth as humans.
However, not all of us evangelicals do so. Some of us (me, for instance) see "original sin" as inheriting a tendency instead of a standing. Without Christ I will choose sinful ways, so I need Christ in me ("Christ in you, the hope of glory," "If anyone is in Christ he is a new creation") to live the life of right, the life of God, here on this earth where I am obviously fallen, and that by choice.
So some--an increasing number, too, I believe, though I do not have the numbers to know that for sure--believe that "original sin" is a moral condition in which I will most of the time choose myself and my self-centered advantage over someone else's good. Once I do that, I have chosen sin and rejected God, and that is when I become one who chooses hell. Hell is chosen, by the way, it is not something that we are sent to by God willy-nilly.
Not so immoral. It takes seriously our own moral freedom and responsibility.