Well, how Hobbesian! Of course, this is not the only possible conception of justice, which is, as Plato put it (in Jowett's translation), "sometimes spoken of as the virtue of an individual, and sometimes as the virtue of a State." But in either case, if we want to get all philosophical about it, I share your skepticism... if justice is a virtue, can we really have a virtue as a service (VaaS)?
Seriously, though, I don't think the OP's point is to propose administration of justice outside the state, but rather that certain tech advances can make it economically feasible for small, individual claims to be given a "fair shake" within the state system; claims which currently are too small to be worth a lawyer's---or even the claimant's---time without said tech.
So objecting to the word "justice" in the context ignores long-established usage of the word.
But out desire for revenge i.e. a negative consequence to those who wronged you, is the fundamental source of our idea for justice, just formalised for a city life.
The guarantee of negative consequence of equal proportion represents a game theoretically optimal approach to discouraging others from harming you.
And if they kill one of yours, then to maintain said deterrence you have to kill one of theirs. As reflected by laws and customs for most of human societies throughout history.