I heard this argument about Rust vs. C that Rust might be memory safe, but the reason why memory safety issues are so prominent in C programs, is that basically every other kind of problem has been fixed throughout its lifetime, so these are the only kind of issues that remain. Both in terms of security and stability.

This is very much not the case for programs that are much newer, even if they are written in Rust they still need years of maturation before they reach the quality of older C programs, as Rust programs suffer from non-memory safety issues just as much. That's why just rewriting things in Rust isn't a panacea.

The perfect example of this the Rust coreutils drama that has been going on.

I don't agree with that assessment at all. The reason memory safety issues are so prominent is that they are extremely likely to be exploitable. Of course you can write exploitable bugs in any language, but most bug classes are unlikely to be exploitable. A bug that always crashes is about a trillion times less severe than a bug that allows someone else to take control of your computer.

I can only quote (from the top of my head) the Android team's findings, that having a C++ codebase extended with Rust cut down significantly on the number of memory safety-related issues. The reasoning was that since the stable C++ codebase was no longer actively changed, only patched, and new features were implemented in Rust, the C++ codebase could go through this stabilization phase where almost all safety issues are found.