My paths argument isn't an argument that you should take a path. It's an argument that taking a path doesn't endorse the existence of the path. When pro-liberty voters boycott elections, that doesn't weaken statism, it strengthens it. No state will ever be overthrown through boycott (i.e. personal secession). Your choices are either flight, revolt, geographic secession, reform, or surrender. You can pretend all you want that what you're doing is akin to revolt, but it looks, walks, and quacks like surrender.
I'm not going to argue that anarchists should vote -- i.e. pursue the reform option. I think any anarchist who actually believes what he says should pursue either flight, revolt, or geographic secession. I don't see how dropping your ballot constitutes one of these three options.
Voting clearly confers an ability to effect change. That your one vote has no more such ability than the votes of millions of others is not something to whine about, because the only alternative to the potential tyranny of the many is the potential tyranny of the few. (This is especially true under the alleged third alternative of anarchy, as demonstrated by 100% of every circumstance in which organized crime has operated. Private markets in justice is perhaps the most thoroughly-refuted experiment in all the social sciences.)
I don't use the phrase "ideological purity" to "smear", because I'm not shy about proclaiming my own ideological purity. In case you somehow missed it, I consider myself to be a more ideologically pure libertarian than any of you confused anarcholibertarians.
The equal right to protect the rights of others is a corollary of the equal right to do anything that is not aggression. So it comes from wherever you think the latter right comes from. Libertarians don't all agree why persons each are endowed with the rights we all recognize, so in a political context I just hold these truths to be self-evident.
Your assertion that the U.S. government enslaves you or me is simply a coarse insult to the memory of every actual slave who died for the dream of even a fraction of the freedoms you take for granted -- a level of secured freedom that is without parallel in the history of our species and planet.
I beg the state for nothing. I just agree with Jefferson that to secure our rights, governments are instituted among Men, and are controlled by Man through representative democracy. Jefferson was no beggar. When ballots were not an option, he chose bullets. If I ever give up on ballots, I'll be reaching for my bullets. Reaching for your keyboard while you drop your ballot does not make you a revolutionary, it just makes you a partially-disarmed reformer.
If you want to be more than an armchair anarchist, then don't just renounce your involvement with the state's mechanism for self-governance -- the one place where the state exposes its neck to you. Instead, renounce your use of all of the state's services -- streets, water, sewage, power, police protection, justice, etc. (Of course, it's difficult or impossible for you to not use certain public goods like protection from contagion, conflagration, foreign invasion, etc.) You yourself just said "When you use something, you validate it, you give it your personal sanction." OK, so practice what you preach. Stop using every service whose provision is predicated on state force. And of course, stop paying any taxes, because to pay them is just to beg the state not to punish you for ignoring its monopolies. Go live in the woods like Ted Kaczynski, only don't endorse the state like he did by occasionally walking along the state's roads to go use the state's postal service. Be a REAL personal secessionist, not like that state-begging wimp Kaczynski.
Or be a revolutionary. Go park a Ryder truckful of ammonium nitrate next to a government building like the heroic Tim McVeigh did. After all, every single person in the Murrah Federal Building was using the State's services, and thus had "validated" the State and given their "personal sanction" to the State making you a "slave". They all thus clearly deserved to die. We need to make such slavery-validating children think twice before they give any more personal sanction to the State by using a government daycare center.
You're young. Your views are evolving rapidly. In time you'll look back at such "slave" rhetoric and wince, just as I wince at rhetoric I once used when I was not much younger than you. But that's OK -- far better to live an examined life and grow from the self-examinations, than to run permanently aground on the first dogma you find that seems to offer safe harbor from the storms of uncertainty about one's available choices.