One of the reasons why Rust is taking over userland is that it's getting more and more difficult to find people willing to maintain a C code base—especially an old one—while there's no shortage of kids willing and eager to hack in Rust.
C will end up in exactly the same place as COBOL: there will be applications that depend on code written in it for decades to come, but they will be maintained by very well-paid grognards simply because those are the only people who know how, and are willing (for a steep price), to maintain them.
Maybe. I think C will continue to have relevance in certain niches like embedded development, and that efforts like Fil-C will quietly carry the torch for a long time. C will also continue to have an important role in bootstrapping.
I generally I don't like the "Rust is killing C" meme because a.) I don't believe it b.) C doesn't need to die for Rust to succeed and c.) it leads to language wars and hard feelings. Rust doesn't really do what C does; it's not a lingua franca among architectures. They compete in some niches and not others.