If the ultimate goal is to avoid backdoors in critical infrastructures (think government systems, financial sector, transportation,...) you could force those organizations to use forks managed by an entity like CISA, NIST or whatever.
If the ultimate goal is to avoid backdoors in random systems (i.e. for "opportunistic attacks"), you have to keep in mind random people and non-critical companies can and will install unknown OSS projects as well as unknown proprietary stuff, known but unmaintained proprietary stuff (think Windows XP), self-maintained code, and so on. Enforcing TS clearances on OSS projects would not significantly mitigate that risk, IMHO.
Not to mention that, as we now know, allies spy and backdoor allies (or at least they try)... so an international alliance doesn't mean intelligence agencies won't try to backdoor systems owned by other countries, even if they are "allies".