The "core systems of Linux" include the Linux kernel, openssh, xz and similar libraries, coreutils, openssl, systemd, dns and ntp clients, possibly curl and wget (what if a GET on a remote system leaks data?),... which are usually separate projects.
The most practical way to establish some uniform governance over how people use those tools would involve a new OS distribution, kinda like Debian, Fedora, Slackware,... but managed by NIST or equivalent, which takes whatever they want from upstream and enrich it with other features.
But it doesn't stop here. What about browsers (think about how browsers protect us from XSS)? What about glibc, major interpreters and compilers? How do you deal with random Chrome or VS Code extensions? Not to mention "smart devices"...
Cybersecurity is not just about backdoors, it is also about patching software, avoiding data leaks or misconfigurations, proper password management, network security and much more.
Relying on trusted, TS cleared personnel for OS development doesn't prevent companies from using 5-years old distros or choosing predictable passwords or exposing critical servers to the Internet.
As the saying goes, security is not a product, it's a mindset.