Right; stl sucks. So its work, but you can make ref-safe containers, even thread-safe ones. We do that; we do audio rendering with audio-chain editing on the fly, with no memory issues. It takes care, more care than other languages. But its far from unsolvable.
And the philosophy of Rust is, what if we encoded that "care" into the language itself? That, to me, is a clear win. It is, to me, good systems language design: codifying decades of hard earned "best practices" into the language semantics itself.
Of course it's possible to write correct C++ code, just like it's possible to write correct assembly code. The point is the extra care required: every piece of code needs to be very carefully authored to ensure it's correct, to avoid the myriad pitfalls.
Or you can just trust the language. And if its not right, or not the way you plan to use it, what then? You're stuck unless the language also permits you to roll your own.
Rust does allow you to implement low-level things in itself, by giving an escape hatch into C/C++-like unsafe code (i.e. risk-of-incorrectness is purely opt-in, rather than always-there).
Examples of things efficiently implemented entirely in the standard library in pure Rust (well, with some calls into the operating system/libc): Vec, the std::vector equivalent. Rc, reference counted pointers (statically restricted to a single thread). Arc, thread-safe reference counted pointers. Mutex. Concurrent queues. Hashmap.