"We've found you to be so good at your job we want you to stop doing that job and do something with an almost mutually exclusive skill set."

"It's difficult to take direction from someone who doesn't know (or can't explore) the problem space as well as I do. I find I'm managing up so much that there seems less and less benefit to the current reporting structure I find myself in."

How often do people complain themselves into a new post?

Absolutely possible. The difference might be that PhDs don't conventionally have the 'downward' path available. They have a lower bound to the career roles they can take up. You may well violate that bound and do more 'line worker' type roles, but that'll get funny looks, might actually be harder to get into with the title (some people might even conceal it for such reasons) and the PhD would have been largely wasted, career-advancement speaking.

For a software developer with a regular STEM MSc, there is more of an upper, less of a lower bound, but I'm not planning on testing out those upper boundaries (like, can you rise to C-level of a large corporation? Sure can, but a PhD makes it much easier where I'm from).

A PhD especially in a technical field gives broad insight to problem solving full stop, or at least someone without severe self-esteem issues should consider that is the bar they've cleared. I have a PhD and wormed myself into tons of low positions, doesn't anyone else want to know how the sausage actually gets made?