There are pieces of software that should be given priority for a rewrite in Rust, but most of C software is never going to be rewritten, because there is simply too much of it.
Therefore, even if C did not have any advantage of its own over Rust, there would still be legacy software to maintain and to extend.
The advantages of C include that sometimes, an embedded processor with a proprietary instruction set is provided by the chipmaker with its own C compiler, which is the only compiler supporting the instruction set; that C is still currently used to write the runtimes of higher-level languages (I'm familiar with OCaml, but it isn't too much of a stretch to imagine that the runtimes of Python, Haskell,… are also written in C).
It goes deeper than that, in a couple of places Rust depends on the C standard: the fixed-layout `#[repr(C)]` structs (without that attribute, the compiler is free to reorder the struct fields; with that attribute, it's laid out the way C would do it), and the `extern "C"` function call ABI. The way to call any other language from Rust, or Rust from any other language, is to go through `extern "C"` functions passing `#[repr(C)]` structs. So even if the C language dies one day, parts of it will live in Rust forever (or as long as the Rust language lives).
I sincerely hope this is not the general attitude of the standards committee. Some of us actually prefer C, and would like to see the language continue to flourish.
