No, removing "a" sounds like broken English. Bad idea.
Without "a", "book" sounds like the name of a person or a place, or that there is more information to follow (somehow), or that "book" is not a physical thing but an abstract concept. All of which goes against my usual understanding of what "book" means.
With "a", I hear that "book" is a thing, and there is one but I don't know which book; only one possible book out of many.
I'm not sure where you found that idea about countable nouns, but that is the exact opposite to the rule of countable nouns. Just count 'em. :)