You can have a Federal system that is administered on a local or state level.
You're creating a strawman Bt. Canada is an excellent example of a system run and administered on the provincial level under a federal mandate. Britain has local NHS Trusts. No one is suggesting that we ignore the needs of local communities.
Sirs, I've not read any serious attorney who suggests that a constitutional amendment is required for universal healthcare any more than it was required for Medicare, Medicaid, GI Bill, Interstate Highways, the war on drugs, etc.