You can't just write Rust in a part of the codebase that's all C/C++. Tools for checking the newly written C/C++ code for issues will still be valuable for a very long time.

You actually can? A Rust-written function that exports a C ABI and calls C ABI functions interops just fine with C. Of course that's all unsafe (unless you're doing pure value-based programming and not calling any foreign code), so you don't get much of a safety gain at the single-function level.