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.
An AGI that is 20% better than humans at everything might be able to recursively improve itself effectively without limit, but it also might not. It took billions of humans millennia to invent things smarter than ourselves even in limited domains; such an AGI - even if it is significantly smarter than humans at everything - might take decades or centuries to produce a similar relative improvement. No doubt it would still change our world in massive ways, but it wouldn't be a singularity.
So it is easier to start from the edges, even for AGI.
AGI would still need unfathomable processing power in order to predict which system would work to everyone's needs.
But what if the proposed solution costs lots of people their jobs?
If we had digital superintelligence then surely it could figure out how to actually provide healthcare to people who meed it, and innovate in treatments. An AGI is a singularity event, in the sense that the transformation it would affect on the state of technology and our lives is rapid and unpredictable. I doubt that our society's hundred-ish year old systems would survive such change, and if they did they would be anachronistic and depressing.