The problem turned out to be an older copy of the app on my disk and macOS getting confused and looking in the wrong place.
I'm adding a Widget to an existing app that doesn't currently have one. I have the current, App Store version in /Applications. I've been running the updated version from Xcode or exporting it to my desktop and running it from there. The Widget itself -- as noted in the question -- runs just fine. But it seems that when looking for the Intent used for the configuration, it prefers to look in /Application rather than the Xcode version. As soon as I deleted the original app, the widget and its settings immediately started working.
Topic:
App & System Services
SubTopic:
Core OS
Tags: