That's just the unclear messaging that the right wing is taking advantage of. I don't have a NYT subscription so I can't see the whole article, but as far as I can tell, there's nothing in that article that mentions actually preventing companies from offering private health insurance--it's just assumed (correctly) that these companies will be operating at a miniscule fraction of their current scale when people can obtain health insurance from a single payer, because few people will pay for private health insurance when there are better, cheaper options.
As I've reiterated repeatedly, this isn't a loss of options. Health insurance companies can still operate and consumers can still pay for private health insurance, they'll simply be competing with a cheaper alternative that doesn't have a perverse incentive to deny care.
I suppose you could make the argument that there will be fewer options because some of these insurance companies will go bankrupt, but that's just capitalism--there's nothing about the current system which guarantees health insurance companies will remain solvent or guarantees a variety of insurance options. In practice there are many places in the U.S. where there's effectively only one health insurance option.