@OneWingedShark
That gets me to where it runs, does nothing, then terminates silently. No popups about missing dll’s. The paths made sense; they were in my AppData\Local\alire\cache\msys64\mingw64\bin folder.
@dmitry-kazakov
The GUI seems to have hung. (Window title is now Dependency Walker (Not Responding).) I tried running it from the command line with the /c option and it terminates silently.
Oops, while I was typing this it came up. There are indeed a lot of DLL’s it cannot find, but at least some of them look like Window API’s:
…while others I’m not sure about:

Lots of found dll’s appear at the bottom of that list.
It looks like it, judging from the help page. (It’s listed as an option under “Catalog format specification > Release Information”.)
They seem to have been installed with alire, but not added to the path. Unfortunately, as I noted above, adding them to the path doesn’t seem to get me far.
I was not aware that there was a mingw-w64-x86_64-ada_language_server. I’ve installed it using pacman and yep, that prints out information when I run it. Now I have to see if I can get the VSCode plugin to use it. If not, then I’ll just run it from the command line.
Do you always hand optimize machine code in a hex editor, as well? 
Speaking seriously: I used to format manually, not that long ago. The gnat compiler badgers one about indentation and spacing, to the point where the number of lines of style advice can make it hard to find the actual errors. I could of course turn that off, but I find the recommended style bearable, and then I discovered the Ada & Spark plugin for VSCode, which is useful for many things, among them the automatic formatting which (mostly) makes gnat happy.