Your RIGHTS, actually, do not ever include a right to violate the rights of others. There is not, EVER, a right to kill/rape/loot or whatever, as these activities ALWAYS violate the rights of another. Society exists to allow people to interact with each other in peaceable ways. Governments exist in order to ensure that EVERYONE’S rights are protected. Otherwise, your argument is headed in the right (correct) direction.
Read my other posts, and you’ll see why I put “right” in quotes in the post you are referring to. There is no word in the English language that has the connotation that I was trying to get across in that post. How would you describe free speech in China? Is the freedom of speech a right that has been denied or a right that is nonexistant in China? If we say that it is a right that is nonexistant, one could then argue that in an anarchy, anything you do is now a right. If you approach the situation from the other side; that there is a predefined set of rights that is either secured by the government or denied, then your reasoning stands. My goal would be for the two definitions to ultimately be the same - there is no law forbidding any action that does not infringe upon the rights of others. It would be the responsibility of the individual to be a moral citizen, not the burden of the government to force everyone into the mold.