Embedded is near and dear to me. ... but let me back up a bit.
My school had _a_ computer for engineering student's use. I think it might have had as much as 4k of core, and was programmed with pinboards and punch cards. The backlog of decks to run was stable at a month or six. So we freshmen learned Fortran the Elbonian way; never touched the computer or even a keypunch. I didn't think very highly of any of it.
Five years out of school, I got trained in timeshare BASIC, but didn't have a project to bill the time to, so I didn't do anything with it.
Seven years out of school, my mechanical part of a major project was done and working, but usable software was two years away, at least. The company had one computer, programmed with punch cards, and was so modern that it had a tape drive, which was a big deal. I tried programming; hated it, got laid off.
Twelve+ years out of school, as a project leader, I needed an embedded motor control. My people could put the hardware together, but programmers were not available except by royal decree. So I learned how to program an 8080, then an 8086, then an 8741, then a few more chips, and gradually became the department's eXtreme Programmer, before anybody called it that. I don't want to say I was self-taught, because that outfit had a cadre of really smart people from whom one could learn a lot merely by getting them talking.
Embedded has changed since then. Chip counts are way down, processors are often just blocks of IP in a huge custom chip, and MBA managers have been sucked into mandating C and its descendants by the siren song of cheap people and free reusable code (neither of which actually exist), code bloat runs rampant, and the world now tolerates systems that have, and need, a reset button, thanks mostly to Billy.
To sort of answer your original question, embedded now requires some mix of programming, electronic hardware design, and systems design, all happening at once. Universities are not equipped to teach anything like that. The other part of the problem, assuming you can find a way to acquire the skills you need, is that the breakthrough ideas often get started in small companies that don't want to pay a wage appropriate to the skillset required to implement the ideas.
... There's the core of a career, maybe. If you can position yourself to be the goto guy who reliably cleans up the messes made by idiot MBAs and inexperienced kids, you might be able to make a living at it.
Mike Halloran
Pembroke Pines, FL, USA