Individual independence is a nonsensical concept in this context because humans are not independent from one another.
Private property as a concept is incompatible with individual independence, because its existence is itself dependent on some form of collective agreement. "Ownership" only exists when a group of people agrees that it does and sees a need to enforce it.
Libertarianism, in practice, is entirely aware of the dependencies between individuals, and what it really is about is tipping the scales in favor of property owners, maximizing the power of the strong over the weak by removing the state's role as an equalizer and instead turning it into a sole enforcer of private property, at which point it is functionally indistinguishable from fascism.