A problem there is that some of the most expensive chronic conditions (CVD, Diabetes) can be substantially improved by preventative care.
Hence: The actor paying for chronic care is the one incentivized to provide great preventative care, separating the two will create incentives with negative patient outcomes and cost for chronic illness.
Another less obvious issue is that by splitting healthcare payment between actors, they are weakened against an increasingly monopolistic healthcare provider market. One key reason US healthcare is expensive today is that insurers are weak vs. healthcare providers - splitting them up would further reduce their ability to negotiate better prices.
While there's some truth to the argument that US healthcare is expensive because it pays for much of the worlds medical research, there is plenty of truth to the argument that the rest of the west gets better prices because their single-payer insurance is a massively strong negotiating party fighting for good prices for patients.