(Looking at this from an American centric point-of-view):
The Czar of health-care in the US today is a brain-worm addled, drug-addicted, vaccine-denying, conspiracy mongering, incompetent jackass. And the overall current administration has shown itself to be hostile to basically anyone who isn't a cis-gendered, white, heterosexual, Christian male.
How many of us really trust these people to make good decisions regarding our health-care? A position that they (or their delegates) would find themselves in if we "nationalize health care".
I think this is a classic example of an idea that sounds good on paper, but doesn't survive contact with reality.
I would imagine individual states would manage their own health services, with the federal government acting as more of a coordinating and standard setting body. At least that's how it works in UK, Spain etc.
There's probably no federal agency less related to the issue of nationalized healthcare than the DHHS.
In the UK, it's operated as trusts separate from day to day government. In Canada it's provincially administered. In Australia, Medicare is a national, tax-funded system with independent statutory authorities overseeing parts of it. Germany, France, Japan have social insurance systems.