They aren’t libertarians. They loosely take libertarian to mean the maintenance of personal liberty via courts, police and army through a monopoly.
You have spotted the fallacy to it. In order to have liberty, you can’t be beholden to a monopoly rights provider and under a social contract you did not agree to.
Some people cannot accept the arguments for private courts, I wouldnt use that as a reason to push them out of the tent. I dont know if I can blame them, since no one seems to agree here on much of anything related to private courts. I’ve seen some wild theories thrown around - such as restitution for all crimes (including murder and rape).
Because its a much bigger concept than the concept of private courts. To believe in public courts over private means that one believes, on some level, that it is acceptable to take money from others, with or without their consent.
It doesn’t mean we can’t co-operate with them, so long as our principles are not compromised, I don’t think anyone would suggest that.
OK, this has taken some time, but I think I can finally declare myself a libertarian (i. e. ancap). Even if I still get questions from critics that I have a hard time answering.