Leviticus is an interesting book of the bible and one that kept me thoroughly entertained during church services when I was too young to follow the sermon.
(e:Paul) asked, "Why not edit the book and just have a whole new book?"
I was totally dismissive of the Old Testament for a while and totally ignored it. But now I appreciate how the entire bible gives great context and insight to the teachings of Jesus and his apostle Paul.
Take for example this story from the gospel of Mark.

A woman who has been bleeding for 12 years is in a crowd surrounding Jesus. She reaches out and touches his cloak and is healed. He asks who touched him and the woman in fear admits that it was her. He says, "Daughter, your faith has healed you. Go in peace and be freed from your suffering."
She approaches in fear because according to Leviticus

WITHOUT the context of Leviticus, we can see that Jesus is compassionate and brings healing to the sick.
WITH the context of Leviticus, we see that he is much more than a compassionate healer. He is revolutionary. He brings healing to a suffering woman despite her "uncleanliness" which is in direct contradiction to the law of the land and in direct contradiction to what everyone in the crowd practices. He offers hope to people who suffer under the law - which was never meant to burden people, but had been twisted in such a way that it did.
I meant to add, it blows me away too how people pick and choose what Leviticus OT laws to follow and what not to follow. If you're really interested in learning more about how different types of Christians deal with OT law, I'd be happy to share it.
Rewriting the bible would decrease it's integrity and be a problem for the Christian community at large. Who decides how to rewrite it? Will all Christians follow the rewrite? Is it the inspired word of God through men if we decide to change it? Will if further splinter a faith community that is already tenuously held together?
From another perspective, many atheists already argue that the bible doesn't have integrity because of its translations over the years. So which atheist view should Christians please?? Neither. We have to be true to ourselves.
I think you're overlooking the fact that there is a huge industry of scholarly works and books that do just what you suggest; explain the bible in context and give new understanding to the word without having to touch rewriting the bible.
On one hand, the most important message of the Bible for me is simple, repent and be saved for the kingdom of God is near (aka, turn to God, live a brand new life, and we'll experience great things in the here and after). Then we can tackle the more difficult things like Leviticus.
But I get what you're saying, that the OT stuff really trips people up so they can't even hear the simple and most important message that Christianity offers. But I still think it's better to leave the ugly in and say, there is the ugly. Because life is ugly and we have to contend with that ugliness, rather than just sweep it away and ignore it. Not to mention, sweeping it away and ignoring it would again bring a lot of criticisms from atheists, skeptics, etc...
I think changing the bible wouldn't stop crazy Christian people from hate mongering. I really don't. Obsessive personalities will always find something in the text to take to some extreme over which they can be divisive and hateful. For me, it's a sin problem - not a bible content problem.
I see what you are saying but why not a new book of christ and let christian teachers explain what it was like to bring context to it, or even just as a separate book of the before christ. Having it altogether lets people and many a hate mongering christian mix it all up.
Its still pretty hard for most people to decipher which OT laws are still reasonable and which are archane. I think, honestly, we live in an isolated place where there are progressive christians and not that being a progressive christian is the norm.
The thing that blows my mind still is how people they decide which of the OT laws are reasonable. Its okay to be hating on the gays but eating hooved animals, shell fish and letting their bleeding wives stay in the village and touch their stuff is acceptable.