We don’t have that.
We have some specific programs like Medicare and Medicaid which are not available to everyone. We have a private marketplace that is available to everyone but does not accomplish everything a government-run healthcare program could. Namely, when the insurance companies are not able to make a profit, they are not obliged to be on the exchange offering competitive prices and when they are able to make a profit there isn’t a public program competing with them to necessitate that their prices remain competitive.
The gap in coverage issue is actually separate from that one IMO (although that is a huge problem as well). Of course subsidies help with that but I don’t think it’s likely that any particular issue with our current system can actually be eliminated purely through increasing subsidies alone.
I am in the camp that doesn’t think single-payer is necessary. A multi-payer system where healthcare funding/insurance through a government mechanism is available for everyone. However, that is not what we currently have.