If both parties agree to use the LLM judge and be bound by what it decides, you don't need a real court.
My cousin has an open-and-shut land case. His father was allotted some barren government land ~50 years ago and they have happily been farming it ever since. But suddenly, ~30 years ago the government decided to build a highway near it. Its value went up. And it dawned on some bureaucrat that nobody had made any money from the initial allotment; an injustice that must be fixed pronto! So they passed an order cancelling the initial allotment, and cousin's family went to court. Now, for almost 30 years, the case has been dragging on through the court. They've been told multiple times: take 25% of the current market value of the land and "sell" it to us. But they insist on doing things the proper way, and have been running to/from the courts. They're on their 5th team of lawyers, afaik. Each new lawyer agrees that it's a clear cut case, but month after month it drags on.
Furthermore, courts are paid for primarily by states. The tech solution would run into the same resource constraint that is already present.
All of this ignores a more fundamental aspect of the legal system, having a chance to have your case heard by a competent authority. Institutional sinews are built by doing the hard things and investing in it.