UPDATE:
I got the configure (and make/make install) to run successfully. On a suggestion from the person handling the bug case I opened, I deleted my existing source base and re-extracted it. I then created a subdirectory for the build, went into that subdirectory and issued this command:
..\configure -prefix C:\QtStatic\qt-everywhere-src-5.12.3 -platform win32-g++ -static -static-runtime -opengl desktop -nomake examples -nomake tests -recheck-all
Then ran mingw32-make.
Both the wiki page on this topic, and the powershell script it references, are flawed. And the build process is a bit fragile: even by creating and using a shadow build, I somehow corrupted my source, causing the OpenGL red herring that was driving me crazy. A fresh extraction was the key, but a couple other points:
do use the -platform switch when running configure
-static and -static-runtime are both needed
when in doubt, use the -recheck-all switch
Anyway, this hopefully closes out this nasty little chapter. Thanks to all involved; I'll be back eventually with some follow-up questions on creating standalone apps.