- "If it is really true to the extent people project it, why is this the case?"
Yes, this is true.
For much more PhD`s in this area are produced than faculty or researcher positions are opened.
- "What do most HEP theoretical doctorates end up doing as profession?"
Applied physics, applied mathematics, engineering (especially math- and science-oriented), software development. Also quantitative finance, military applications, what not.
- There is no rule without exception.
At this juncture, and possibly for several more years to come (with no warranty given), one subarea close to HEP may be granting its PhD`s a better chance of employment: Quantum Computing. Not very exactly HEP, it certainly is a topic emerging from QM and, in some applications, overlapping with QFT.
Figuratively speaking, this is a bandwagon. It is now attracting enthusiasm and investment, both from governments and industry. So, if you are finishing a BS or MS in physics, cogitate on this option.
As a possible starting point, I saw favourable reports on this book.