The states have a lot more power than in a lot of federal systems, and the geographic and demographic issues of, say, Florida and Wyoming are likely to be so different that trying to shoehorn both into a federally managed system would be doomed to failure. With federal subsidies the states could spend it how they needed to.
State power is basically irrelevant, and the same with the geographic and demographic issues of different states. The only really
plausible model for something happening in the 1930s or 1940s is a universal
insurance scheme, which means that the federal (and possibly state) governments would simply be paying for people's hospital stays, doctor's visits, etc., but not actually running the hospitals or moving the doctors around or anything like that. Thus, the "geographic and demographic issues," as you put them, would work themselves out more or less naturally.
You can't force the states to do that though, not easily. I guess you could do something like saying "unless you fund UHC we'll not give you any money for Social Security" or something, but I feel that SCOTUS might throw that out.
It would be quite easy, as a matter of fact. Medicaid--which is basically federally and state funded insurance for poor people--was...not
easily, exactly, but relatively straightforwardly accepted by every state in the 1960s. I don't foresee states passing up participation in a similar scheme in the 1930s or 1940s, either. For that matter, Medicare--federally funded insurance for the elderly, mostly--has not been controversial or unsuccessful, either.
Essentially, all this challenge calls for is the establishment of something like Medicaid or Medicare that covers everyone, instead of just the poor/elderly, respectively. It's not that much of a lift to get states to accept it, and the size and federal nature of the country isn't going to be a big problem, looking at those systems.