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.

The magic is in the initial diagnosis being written down, saved and locked.

It's trivial to analyse the pre/post AI involvement doctor diagnosis manually and see what's going on.

If a doctor is just putting "asdljasdaskjd" on the initial to unlock the AI answer, they should be promptly fired.

Then the claim may be that you don't need that many doctors anymore and that one doctor can do the job of X doctors in less time which has the economical effect that there is less demand for/supply of doctors, which then results in a home grown shortage of doctors, since less people are incentivized to become doctors...

Step 2 prevents that. It's not there by accident.

They need to write down their (initial) diagnosis before the AI answer is shown.