Setting ethical debate aside, this has an easy answer. For months many dismissed seriousness/need for distancing/need for masks, citing Trump. Reporting that Trump said privately how deadly COVID is/that it spread airborne could have changed those people's behavior & saved lives https://twitter.com/ErikWemple/status/1303826136384901120

When the president of the United States is publicly underplaying the seriousness of a global pandemic and at times encouraging people to go to rallies/reopen businesses, while only privately acknowledging how deadly the virus is, the potential urgency to report should be obvious

How might reporting save lives? There are Americans who believe the president and thus didn't take the virus seriously. Reporting that he privately admitted to misleading Americans about the seriousness could have reached those Americans, and helped them make healthier choices

reporting what the president was being briefed on/said in private also could have created pressure for him/other admin officials to acknowledge that seriousness in public sooner -- again potentially prompting more people to take things more seriously more quickly & saving lives

Woodward's best defense is that folks spoke explicitly for a book, thus he didn't feel comfortable using the reporting in other contexts. An ethical debate worth having. But no need to be incredulous about idea that reporting sooner could have saved lives- it obviously could have