What makes you think that public school bible classes wouldn't be taught from those same "atheist" websites you seem to think we all go to for information? Public schools would not be able to prohibit Atheist teachers from teaching those classes.
Why would it have to be a school teacher. I bet there'd be all kind of volunteers that would step up for that. As long as the syllabus were made avaiable and parents had enough information to make an informed choice as to whether they wanted their kids to participate, I don't see a problem with it as an extracurricular activity.
...and I take issue with your characterization of atheist sincerity when it comes to having read various religious texts. Survey after survey shows that American Atheists score higher on religious knowledge tests than do their American Christian counterparts...whatever it is that we are doing is working better than what Christians themselves do.
Perfect people don't go to church. Sinners of all types go. And just like any other group, some will take their work (becoming mature in the faith, in this case) more seriously than others.
I would ask you not to dismiss us out of hand like this. When I say I have read the Bible, I mean it...I have read two different "modern" English translations (cover to cover, really), and attempted to get through the KJV...but I couldn't get past the language and writing style. I have also read the Book of Mormon, and the Qur'an (English, obvs).
For many of us it is the Bible itself that pushes us away from Christianity.
Reading the Bible and understanding what it's trying to teach is tough--on your own. It's easy to pick out verses and their apparent discrepancies and hold them up as proof of christians being hypocrites. That's just rote memory and a reasonably quick intellect. I can't tell you how many passages I've read and thought "what the heck is that supposed to mean? It says 'this' in another book" and have had to seek out help to help resolve those conflicts. I remember the first time I read Job and then when I got to study it a bit with some others--man, was my reading of it way off. Studying in groups helps--A BUNCH.