Americans pay twice as much as any other OECD country, yet we are the only country who cannot cover everyone under our current system. By covering everyone, and requiring taxes from everyone to pay for it, we can both help control medical cost inflation and reduce the average cost of health coverage for everyone.
So, yes, this system will require taxes to be collected, but because of a lower overall health system cost it means that you no longer have to pay for: personal insurance premium, employer contributions to employee premiums, workers' compensation premiums, automobile insurance premium component that covers medical claims, cost-shifted insurance premiums to cover the uncompensated claims of the uninsured/underinsured, Medicaid taxes, and more.
By eliminating the glaring inefficiencies in financing health care today, we greatly reduce administrative costs. Also, by pooling all of our money into a common public trust to finance health care needs (a public insurer), we eliminate nearly 30% of today's cost that is completely wasted on private insurance overhead.