The offensive organization wold probably then still sit on vulnerabilities only it knew about, but at least this would be better than the current situation.
It's a myth far as I know. I've studied them a long time seeing much conflicting info about this. A declassified, historical document I found at one point about them said their job was SIGINT and COMSEC (just communications security!) for U.S. government. A later provision extended this to protecting COMSEC of defense contractors. The IAD seems to have policy-driven stuff about helping protect INFOSEC in general. There could've been a COTS mandate of some sort at some point but it was clearly toothless.
The NSA is mandated to protect communication security of defense sector. That's it. Even then, the defense sector keeps asking them to downgrade the security to let in more quick-moving products from commercial sector that are hacker fodder. They've since started on a program that lets them in after a 90-day evaluation against the lowest standards from Common Criteria. The NSA is the last group that should be responsible for INFOSEC given all this w/ market an utter failure, too.
The groups that have done the most are probably NSF and DARPA for funding strong security with NIST and DISA (esp STIGS) at least trying to do something with hardening guides and crypto recommendations. I prefer reputation-driven nonprofits that are funded with combo of donations, licensing of quality software, and consulting fees. They can't get acquired or be destroyed by changes in government policy.
https://twitter.com/Snowden/status/839168025517522944
Maybe if they were required by statute to accept anonymous submissions and make FOIA-style responsible disclosures after a reasonably short period of time, they wouldn't end up colluding right away.