When I was 12, I learned 6502 assembly language. The computer I had did not have a C compiler, only BASIC or assembler (you could also enter hex into memory locations, which was hardcore). Learning assembly is actually a LOT easier than people make out. Assembly is basically mnemonics for the underlying machine code, plus named locations. Assembly has everything a language needs for Turing completeness. Assign, Add, Subtract, Compare, Branch. And THAT is how the computer works 'under the hood'. C is one level of abstraction above that, and while it's true that a LOT of software has been historically written in C, before C came along, most software was written in assembler or BASIC (which is older than C, for the historians).
> Learning C is almost mandatory if you want to see how deep the rabbit hole is.
Not really. I know what is going on 'down there', regardless of which language was used to compile the machine code. What is REALLY useful is to have really succinct, powerful, high-level abstractions to build software quickly and with as little code as possible.
> Knowing it will certainly help when exploring the intricacies of processors and operating systems.
If that's what you want to do, then fine. I'm too much of a utilitarian for that kind of exploration. I want to build stuff.
I think if you want to understand what's going on 'down there' in the 'rabbit hole', learn assembly language. There are only seven to ten basic instructions and you can learn them in a couple of days. You can build anything you want in assembly language, if you have an eternity to do it in.