Why? That idea is rather silly IMHO.
You have to ask yourself what purpose the insurance companies serve? Why do they exist?
The simple answer is that they exist to establish risk. An insurance companies sole purpose is to establish what your overall risk is to determine what you need to pay into the whole in order to come out even at worst. That is it. They do not exist to pay your bills - that is the consequence of the system not its primary purpose. If the government is paying those premiums then what purpose does the insurance companies serve? Nothing at all. They would make money for essentially moving money around.
You're conflating an insurance company with an
actuary. Obviously insurers will employ actuaries for the function you're describing but more broadly insurers
pool risk. They design and sell a financial product, the point being that they take on risk for your expenses if you have a health issue. (This is unavoidable--health spending is incredibly concentrated on a relative small segment of the population at any given time, which is why an insurance model--whether public or private--will always be needed to finance it.) And in doing so they have to do all of the administrative side of that: building and maintaining provider networks, negotiating prices, and yes actually paying the claims when you have a health issue.
If you imagine the basic template for single-payer health care being essentially Medicare-for-all, that program is still going to be administered by private insurers. That's how Medicare has always worked (today those insurers are called
Medicare Administrative Contractors). They pay the claims, even as the government bears the actual financial risk and sets prices.
However, the trend of the past 20 years or so in public insurance has been away from that model. State Medicaid programs have gotten away from it and gone toward risk-bearing managed care organizations. In other words,
they privatized. People in Medicaid tend to enroll in private insurance plans these days and their insurers are at risk for their expenses, not the state.
Same thing for the subset of Medicare that has been privatized (Medicare Advantage). Enrollees get to pick whether they want to enroll in traditional Medicare (which, again is still actually administered in practice by the MACs) or enroll in Medicare Advantage where they choose a private insurance plan. Enrollment in the latter has been growing pretty quickly.
So I would not expect that even if we ultimately migrate toward a "single-payer" model that it wouldn't follow the same trends that have re-shaped the single-payer programs we've had since the '60s. Namely partially privatized with less risk for the public sector and more risk for the private sector (in addition to the inevitable administrative role for the private sector).
