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Why would we spiral into chaos? Is that what hollywood told you?
If people were actually wise enough to come to the conclusion that voting is nothing but giving permission to some assholes to enslave you, then they'd be wise enough to form a completely voluntary society based on the private sector & free market.
This is what Anarcho-Capitalism is all about. We know that if you just deleted those in power now a vacuum would form and the useful idiots would all just demand to be ruled by somebody new. That's why it takes education first so they don't want that anymore.
So I don't see it spiraling into chaos by any means. If they somehow voted all 'the bums out' then they'd simply want new bums, and hold emergency elections to get them.
But if it's like you say, everyone refused to vote, then they came to the (obvious, IMHO) conclusion that voting is bad because having rulers is bad. This would be every anarchist's dream come true.
Civilization would finally begin to flourish in a way that you can't even conceive right now. No regulations, no bureaucracy, no bad laws, no bad cops, nothing at all to make life harder on you than it would be in nature.
For those looking to learn more about this extremely-well-explored field of thought, I recommend starting with Larkin Rose videos on YouTube, and then graduating to of the many many many books on anarchocapitalism at Mises.org, like the Machinery of Freedom or basically anything written by Rothbard.