"Just looking at daily data" is your mistake. Essentially, it takes N days for the virus to kill, so deaths will lag confirmed cases by N days, and thus the calculated case fatality rate using this method is incorrect. This is why epidemiologists call it a "naïve esimator" (see section 3.1 in https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4540071/).