Under (pure) Capitalism, there isn't public spending for private individuals. Not to mention government influenced pricing... single payer is pretty far away from laissez-faire and capitalism.

Capitalism is simply an economic system based on private ownership of property. Public spending is completely irrelevant. Single payer systems are entirely compatible with capitalism, and is a good example of Social Democratic policy (not to be confused with Socialism, which it is not), which seeks to work within the structure of Capitalism. A Socialist system would be one in which private ownership (of health care) is prohibited; perhaps the closest example among modern democracies being the NHS in the UK.