I am not in favor of shuttering one of the agencies responsible for public health, although I can see why my comment could be read either way. If you believe in reform there are compassionate ways to do it.
CDC probably does more public health? The work NIH funds through its grant process is not broadly "public health"; it's virtually all American biotechnology research and bio/biochem basic research. Public health deals with the interface of health knowledge and public policy and communication; people credentialed in public health are not, generally, practicing scientists as we'd think about them. NIH grant researchers, on the other hand, are wearing white lab coats and working with fume hoods.
No, HHS does the vast majority of public health research in the US, which includes both CDC and NIH. The CDC budget is roughly a quarter of the NIH budget; most public health research is NIH-funded. The CDC is more operational- they take what is learned and apply it to prevention and treatment.
It may in fact be the case that more HHS-funded public health work is downstream of NIH than of CDC, but NIH is obviously not primarily a public health organization; here's the most recent RePORT data:
