I bought the CPU from Newegg. It did not appear to have been used. The packaging was sealed and tight. The thermal paste on the cpu fan was clean and the CPU showed no evidence of having had thermal paste applied and cleaned.
I have had a few DOA components over the years, but never a CPU. It seems a slim chance that it was defective out of the box. Newegg will take it back no questions asked, but it would REALLY suck to wait several more days for new one to arrive only to find the issue unchanged.
This page on the ASRock website indicates the AB350M Pro4 board supports the 3400G APU:
https://www.asrock.com/MB/AMD/AB350M Pro4/index.asp#CPU
However, looking more closely at the BIOS versions page here:
https://www.asrock.com/MB/AMD/AB350M Pro4/index.asp#BIOS , I see a potential issue. I updated to the latest version, v.6.60. It does not reference the need for the All In One driver that other versions do, but it does say "read the description for the previous BIOS version". So does the previous version. If you follow that chain far enough back, you eventually get to v5.90, which indicates the need to install the All In One Driver. And to do to it BEFORE updating BIOS. It includes instructions for how to roll back, if necessary. I think I'll try that, then re-up to v6.60 and see what happens.
Thanks again for all your help!
[ EDIT ] - Reverting to an older BIOS and installing the All In One driver, then re-flashing to the v6.60 BIOS did not solve the problem, either. Doesn't even look to be
thinking about posting. No peripherals like keyboard and mouse are getting any power. Live-plugging into the onboard HDMI doesn't even make the screen flicker. The case LED's light up and all the fans spin, but otherwise there's no indication that it's anything other than a brick. Either the CPU is bad, ASRock's website is inaccurate or something I can't explain has gone wrong.