Maybe it's not worth learning C as a programming language, but it's worth learning the C standard library, as every language links it in and most of the functionality is available. I'd also say to throw in a cursory reading of the Linux kernel's syscalls as exposed through glibc and a tutorial on how to go from glibc to the kernel implementation of the syscall. It's not strictly necessary for game development or frontend website design, but in games you'll end up learning C++ anyway and with websites everyone seems to migrate into the backend servers eventually or else quit programming and do graphics design.

As far as operating systems, Rust does seem to be a better entry point, with the Redox OS and general community.