But why this? What's the purpose of prohibiting private health insurers, and what's it trying to emulate
As I said earlier in-thread: What's the point of having them? Where is the circumstance in which having "a choice of options" would be preferable to "everything is covered and you don't have to worry about paying for it."
Sure you have your Tims rallying the banners around elective non-medically necessary cosmetic surgeries and your braces and the like. But those sorts of services aren't explicitly prohibited in the ban. In fact that's precisely at the heart of the bill. If there's something that M4A doesn't cover that you want to do, then that is something not covered by M4A and therefore can be covered by private supplemental insurance.
As for "the purpose" other than what I above elaborated:
a) For-profit healthcare is quite clearly immoral. In a capitalist system the sole driving motive is the pursuit of profit, and therefore, a for-profit Healthcare provider's interests
can never be aligned with the interests of the client, because providing healthcare is something which cuts into profits, so the incentive will always be to mitigate the times in which healthcare is provided.
b) Leaving aside the self-evident point of morality, the power of UHC from an economic standpoint is in the collective bargaining. UHC drives down costs because the single-payer is negotiating costs on behalf of a big tent. "Give us this service at y cost or you will not give this service at all." Part of the reason health care is so fudged in the US is because the tent is split up among a number of providers making it difficult to negotiate for lower prices. The consequence of Pfizer selling their Alzheimer's drug at $700 a pill isn't that Pfizer doesn't get to sell their pill, it's that (maybe) Blue Shield doesn't cover the pill, but Kaiser et al
will. Also another big economic argument for single-payer is that much of a Hospital's cost comes down to administrative/billing costs associated with having to have departments to deal with every healthcare provider a given hospital covers. Not eliminating or at least
significantly reducing the position of the Private Healthcare Industry undermines this advantage. It instead turns into the
xkcd comic about Standards
c) Consider schools. The presence of private schools and charters actively prevent our educational system from improving broadly. We have created a divided system whereby those with money and power can direct that money and power towards the maintenance of high quality schools for themselves and their children, and away from the public school system, to the detriment of everybody else. Why should the wealthy care when they have no "skin in the game" as it were? The result would be the same here. It's a lot easier for the government to debate kneecapping Medicare funding when they and their wealthy donors are already insured by their premium plans and don't have to worry about the consequences to their own well-being. In a single-payer system, everybody is invested in the system being good because its not-goodness would directly impact their own quality-of-life and outcomes.