I disagree, although even if there is a C interface, it can still be made to work with many other programming languages too, so if you prefer Rust (or Python, etc) instead, then you can use Rust and you do not have to use C. Either way, it allows native code will full system capabilities, which an end user might want in a browser extension.

(JavaScript-based browser extensions are still useful to make simple things without writing as much code, and also useful when you want to distribute portable sandboxed code for browser extensions, but for many other uses, C will be better.)

(It is true the C programming language has some problems, but so do all programming languages.)