Would a single-payer system be less constitutionally problematic than a mandate?

[anvi]

I don't advocate single-payer systems, being more a fan of the Bismarck system.  But I don't see any reason why a single-payer system, such as exists in the U.K. and Canada (though technically the Canadian system is not a "single-"payer system), would breech the U.S. constitution.  In such systems, the government simply pays medical expenses incurred from patient treatment, and finances the payments through taxation.  The constitutional controversy in the U.S. seems to be over whether the government can mandate the purchase of health insurance of its citizens.  No such mandates exist in "single-payer" systems; instead, governments levy taxes, which they are of course constitutionally empowered to do, and then pay for medical bills with those revenues.  

However, in Bismarck systems, by contrast, private (non-profit) health insurance companies pay for medical bills, and in these systems, individual mandates to purchase insurance do exist, as they must, so that universal coverage can be financed by those companies.  ("Single-payer" systems have private health insurance companies too, but only to pay for services that the government may not.)