> Yet I'd be very hard-pressed to tell you how exactly CPU instructions work, and have never touched anything "closer to the metal" than C. Should this disqualify me from the profession?
No - it shouldn't. I feel that you should want to know this level and more about the internals (and history) of computers. I'm biased in that has been a driver for me personally...
...but to give you a hint:
A CPU (in the most basic sense) is nothing more than a very fast and very complex player piano, regulated by a clock. At each "tick" of that clock a row of "holes" (bits) representing a variety of information (address, data, instruction) is presented to the CPU. These bits are actually voltage levels. The CPU uses these voltage levels to change the state of logic gates in the system (which are wired for a variety of different tasks - but you only need one particular kind of gate to make any others - a NAND or NOR), which determine what happens, what to do next, etc.
I won't go further - and let me be clear that what I wrote above is nowhere near complete, and only describes the most basic level of an "ideal" CPU (or ALU or whatever) - because in the real world, you have things like voltage transients, timing issues (signals not arriving at the same times - hence things like wait states and such), noise, plus things like triggering on rising or falling edges of clock signals, multiple clocks staggered in time (for multiple triggerings per "cycle"), and a whole host of other things.
Yeah - it's pretty complex. But at the heart of it, it is pretty simple - a player piano, more or less, with a music reel spinning at mind-boggling speeds. If you need or want more information, there are (like everything else) plenty of resources on the internet about how all this stuff works. Check out some of the sites on people who have built computers from relays or logic gates; they typically have a lot of good information about the lower level stuff. Also look into some of the old books on building computers, and how they work (I'm talking the practical computing books from the 1950s and 1960s - difficult to find at times, check on used books online, in stores, plus on google books and archive.org; also, old back issues of Byte Magazine from the 70s and 80s are a good resource).