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I guarantee in their mind they've created a rationalization devoid of logic

That's the thing, they're not actually seeking to live on their own terms. They're not radical libertarians.

They believe the US government was replaced by a corporation by Lincoln, and birth certificates are actually representative of stock in said corporation backed by the US Treasury that's traded on the stock market, and that if you just know to ask you can cash out your secret corporate account, which is what your name in all uppercase letters, as found on your birth certificate, represents.
Every interaction with the false government is actually mediated by Byzantine rules, but you can always opt out, since they're a corporation and have to play by the rules.

In their minds, they're not seeking to live free, they're checking "do not agree" on the TOS to driving a car, which means they don't have to follow the rules because they didn't sign the contract.
Same as telling Comcast you didn't want phone service, but you'll accept one month free internet, they don't feel interested in signing the "speeding ticket" contract, but they would be happy to cash out their secret bank account.

Say what you want about the viability of extreme libertarianism, "I don't want the government telling me what to do" is a coherent stance.
Sovereign citizens are just unhinged, usually because of life stresses and a helping hand from someone who wanted to make a quick buck. They still dangerous, despite often being victims themselves, however.