A more realistic step 7 is that physicians gradually align their diagnoses with the LLM as they sacrifice to Moloch in order to (temporarily) game the metric. Eventually the humans become little more than an imperfect proxy for the LLMs and are eliminated.

I agree with GP's solution but we'd need regulation to prohibit what you describe.

Why would private equity want more competent doctors?

Incompetent ones order unnecessary tests and exhaust treatment possibilities, which drives up cost billed to insurance.

Only the insurance industry and perhaps licensing bodies can pressure to keep the quality floor high, at least in terms of accurate diagnosis and prevention of overtreatment.