What were those budgets doing before?
For one thing, we already pay for the uninsured. Those peo0ple who go to the emergency room for the stomach aches that were being disparaged before are still paid for. The government has a fund that gets paid to hospitals to cover the expense of un/underinsured people using the emergency room. If we covered those people adequately, we could eliminate that fund, use it to pay for insurance, the poeple could go get check-ups and combat the underlying problems rather than waiting until it's so bad they need to go to the emergency room, thus freeing up the space for real emergencies and spending less per person on the program.
I agree that a single-payer program is the best. If we took the money being spent on current insureres and diverted it to a tax, I'd pay exactly the same, but now I could go to any hospital, whether I'm on vacation across the country or not, and everyone could get the checkups and tests they need to maintain health.