Functional languages are inherently built on non-functional ones, for the same reason that object oriented languages are built on non object oriented ones, because cpus are fundamentally not object oriented or functional.

They are computational machines with specific instructions around moving values in memory and performing certain operations on them.

Assembly / machine code is always, at a fundamental level, procedural programming because at a the most fundamental level, cpus are designed as procedural machines, so all higher ordered languages, from C, all the way up through Python and Lisp, have to translate down to assembly, which maps to the machine's instruction set, which is procedural and imperative.

However, there is an OS that tries to be functional as much as possible though, and that's a Linux distribution called NixOS, based on the functional language Nix.

Emacs :)

Ok joking aside there were these things: https://en.wikipedia.org/wiki/Lisp_machine

There also are modern examples but they tend to be for specific niches like Mirage.

A few good reasons, the first being that brand new operating systems don't get written all that often. But even if they were, functional languages focus a lot on abstractions, making them generally higher level languages and so not fast enough to compete with C.

Having said that, Rust's design is quite inspired by functional languages in many ways, and it is indeed being used in operating systems.

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.