I'd say it's plenty possible. The mission of the NSA is collecting information and providing that information to the government and military. They are not in the business of defending infrastructure, whether public or private. It makes perfect sense that they would collect information about back doors they could use later against identified targets.
"NSA's mission is to help protect national security by providing policy makers and military commanders with the intelligence information they need to do their jobs."
For the NSA to hand over information about identified back doors in Windows to Microsoft directly would be clearly outside this mission. That said, the NSA should have informed someone in the government of these back doors, at least in a general sense, so policy makers could have decided whether it would have been better to convey it to Microsoft or to let the NSA continue to keep them secret. Maybe they did. Who knows?
To the extent their mission doesn't involve making us more secure, why are my taxes paying for them? The NSA sounds like a swamp that needs draining, in the parlance of our times. They're breeding mosquitoes.
More about the NSA's mission here: https://www.nsa.gov/news-features/press-room/statements/2013...
"NSA's mission is to help protect national security by providing policy makers and military commanders with the intelligence information they need to do their jobs."
For the NSA to hand over information about identified back doors in Windows to Microsoft directly would be clearly outside this mission. That said, the NSA should have informed someone in the government of these back doors, at least in a general sense, so policy makers could have decided whether it would have been better to convey it to Microsoft or to let the NSA continue to keep them secret. Maybe they did. Who knows?