The trade off is between measurement bias and selection bias. Typically you want to be dealing with measurement bias and not selection bias, since the latter is often impossible to account for.

You are correct that it is preferable to report the raw data, but it is more than just "honest"! Even with that data `incorrectly` on the wrong day, when binned as a full month the growth rate will be correct. That is, the data is now more correct than if the late reported deaths were not included.

I believe this is effectively the same as Intention to Treat Analysis (https://en.wikipedia.org/wiki/Intention-to-treat_analysis).