Other embedded programmers are just that, programmers who found a programming job that happens to be in this sector and learned the peculiriaties on-job.
Also bear in mind that there are different levels of "embedded" which ranges from writing code for some obscure MCU directly in hexeditor (as it is so obscure so nobody bothered to write assembler for it) to writing applications in Java for something that is essentially android tablet bolted onto some larger system (be it car, airplane or some industrial machinery)
Edit: also habing custom hardware is not that capital intensive unless you plan to mass manufacture said custom hardware. And almost any piece of custom hardware has some MCU programmed in C and/or device driver which is also written in C.
Over last 10 years I've participated on development of about 5 different pieces of custom hardware for various niche applications. This includes industrial sensors, IoT-ish stuff and somewhat peculiar reliability and security enhanced PC platform.
Math has the nature of simplicity -- well defined and little complication, but abstract (or useless).
If you like building things, I would suggest a text editor or a GUI layout engine. For the latter, think about an intuitive declarative way of describing layout (like HTML/CSS or TeX or Tk) then implement it on native win32 API or GTK. Focus on the simplicity and in-control aspect; do not get distracted by feature completeness.
Both of these sound like great learning projects - but are they too far in the "deep end" for a beginner?
Just curious :)
If you're OK with the dated context it's still applicable to modern Win32. You can find it used.
I always think of people who use C professionally as people who have been doing it for a _very_ long time and know a lot about it. In that situation it would be exceptionally hard to find any employment working with C.
I've always been hesitant at many Project Euler problems since some of the math is off putting to me. That's why I consider building something (like a side project or something of use).
Using C as a learning tool to better understand programming would be a great idea and I'll likely venture into that- hopefully something comes of it though! :)