You could counter that health insurers will just start staffing their call centers with their own robot army, but I suspect that the occasional incredibly expensive hallucination will put an end to that at least in the medium term.
What it can do is create the best simulation and a model representation and bruteforce on top of that and then take actions, but the model will still be far from perfect and it likely just will help do a bit wiser or optimal decisions, that are like 10 percent efficiency gain here, 20 percent there, etc. And it will take years to see results.
This is actually one of the areas where I'm pretty optimistic about LLMs (at least if used defensively against, not aggressively by, sprawling bureaucracies).
LLMs can be the email-to-dozens-of-hours-on-the-phone-for-personal-reasons adapter anyone other than billionaires and company execs could only dream of until now.
But if you can provide statistics that a given health insurer is practically not answering 90% or so of their calls (because they time out at the phone level), that's probably not something they can ignore in the long term.