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If that's true, then they're the worst kind of private, the kind where there's no competition. So technically private yes, but in the most abhorrent and corrupt way. The libertarian position is that such links must be eradicated.

For a lot of things, that eradication can be done by simply removing the whole thing (e.g. how Milei seems to be doing in Argentina), but those three things most probably cannot exist privately. Thing is, the fewer things we as citizens have to be very vigilant about (and I dare to say that everyone agrees these three are such things -- some people just argue that they are not the only things), the easier it is to do so.