There a many, many jobs in the health insurance industry in this country. Would we simply just eliminate that industry, or would we organize our plan through the industry somehow? I kind of doubt the latter, because the folks who want a single-payer system generally want every single person to be covered, viewing health care as an entitlement due to all citizens, not something that is simply "available." Trying to do it through subsidized premiums, as you mentioned, means that you'd have to elect to sign up for something in the first place. I don't think that's what the preferred single-payer system would require.
Canada has single-payer but there are private options available to those who can afford it. I have a few colleagues who do the basic stuff via the public avenues, but if they want to see a specialist for a unique situation, they'll go the private route. I suspect that's a common thing.