Some parts of public health can be handled locally, and some parts of public health can only be handled on a national or international level. They can't figure out the pattern of an epidemic based on local occurrences alone.
State health departments don't have the equipment and expertise to do a lot of things. Hospital disease laboratories are only equipped to identify infections that are common in their area. Why stock a laboratory with expensive agents that you'll never use? When hospitals get a patient with an unusual disease, they can't identify it in their own labs and they have to send the samples to the CDC.
That part I agree with - pattern analysis and research are definitely in the purview of higher levels.
You can't give somebody a responsibility without giving them the resources (financial and otherwise).
That part, however, I don't: where do you think that money comes from in the first place? It makes no sense to send money to a central location just to beg for it back (except for emergencies that overwhelm the local region). The day to day stuff should be funded locally.
Although it's a different topic, the same goes for transportation funding: it makes no sense for Portland to pay for Boston's big dig and Boston to pay for Portland's light rail, and Wyoming shouldn't have to pay for either. There's a case to be made for the reverse: helping rural areas with transportation needs, though only to a limited extent.