Really, you should have ended right here:
"...Fifth Amendment lets you refuse to answer the question of whether you even committed the crime at all, and I didn't see what was so great about that, because it is everybody's legitimate business whether or not you committed the crime."
Here's why: You don't have to say shit to anyone about anything who is accusing you of a crime. The onus is completely on them to provide the evidence. They can ask a judge for warrants to search your person or property, but NO ONE can compel you to answer any questions as long as you tell them that you invoke your 5th amendment privileges and refuse to answer *any and all* questions. Once you begin answering questions you open yourself up to having anything you say used against you in a court of law, and NO ONE wants that.