...which is exactly what they do. Which, I'm sure, you know.
The thing is, that "burden" includes flu-induced pneumonia and other conditions that are worsened/exacerbated due to flu, or that are opportunistic due to flu-caused immune-system weakening.
That may sound similar to what has been done for COVID; however, the difference is that flu deaths don't subsume reported pneumonia deaths - whereas currently, COVID deaths completely subsume flu deaths, pneumonia deaths, and all manner of co-morbidities and opportunistic diseases/disorders.
Which also dilutes the "only the flu" argument.
We actually don't know what "the flu" does in any given year, with any accuracy. We have relied on... wait for it... modeling.
ETA:
In a way, this constitutes a defense for how the CDC has handled COVID reporting. They've never (as far as I can tell) been asked to collate data on actual deaths from anything (other than guns) in a comprehensive or meaningful way. Hard to expect them to do it for something new.