Not a system programmer -- at this point, does C hold any significant advantage over Rust? Is it inevitable that everything written in C is going to be gradually converted to safer languages?
There will be small niches leftover:
* Embedded - This will always be C. No memory allocation means no Rust benefits. Rust is also too complex for smaller systems to write compilers.
* OS / Kernel - Nearly all of the relevant code is unsafe. There aren't many real benefits. It will happen anyways due to grant funding requirements. This will take decades, maybe a century. A better alternative would be a verified kernel with formal methods and a Linux compatibility layer, but that is pie in the sky.
* Game Engines - Rust screwed up its standard library by not putting custom allocation at the center of it. Until we get a Rust version of the EASTL, adoption will be slow at best.
* High Frequency Traders - They would care about the standard library except they are moving on from C++ to VHDL for their time-sensitive stuff. I would bet they move to a garbage-collected language for everything else, either Java or Go.
* Browsers - Despite being born in a browser, Rust is unlikely to make any inroads. Mozilla lost their ability to make effective change and already killed their Rust project once. Google has probably the largest C++ codebase in the world. Migrating to Rust would be so expensive that the board would squash it.
* High-Throughput Services - This is where I see the bulk of Rust adoption. I would be surprised if major rewrites aren't already underway.