There can be, although some parts may still need to be written in assembly (which is imperative, because that's ultimately what most CPUs do), for parts like a kernel's context switching logic. But C has similar restrictions, like how it is impossible to start a C function without initializing the stack. Exception: some CPUs (eg Cortex M) have a specialized mechanism to initialize the stack.

As for why C, it's a low-level language that maps well to most CPU's native assembly language. If instead we had stack-based CPUs -- eg Lisp Machines or a real Java Machine -- then we'd probably be using other languages to write an OS for those systems.