Q:  If the Bible was allowed in public schools, wouldn't this be promoting a religion?  Isn't public education supposed to be neutral?

This is one of those many places where some of my friends, in this case those at Answers in Genesis have got it exactly  right. I defer now to their answer:

A:  From a Biblical perspective there's no such thing as neutrality. 
Scripture says that you either walk in light or darkness--you're either for Christ or against Him--there's NO in-between.

When the humanists threw the Bible out of the public schools, they simply replaced it with the religion of humanism.  Just because public education doesn't mention God doesn't mean it's neutral.  In fact, the philosophy that's being taught to students now is that they can think, believe, and act apart from God.  That's the religion of humanism.

Children are being told they can determine truth apart from God--that
fallible man is the authority.  Thus even Christian children who are in this system come out thinking like humanists.  When they get into positions of authority, such as in government, they don't make correct decisions based on God's Word.  They tend to think that the Bible is simply an interesting ‘guide book’ to life that you somehow add to your thinking.

We need to build all our thinking on God's Word--starting right from Genesis.