Thanks @JohnAtl — great tip. That indeed might have helped (not sure if that window would have an identifiable parent, as the app in question is headless so selecting the window doesn’t give focus to a specific app / change the menu bar / etc.), but I believe I solved the mystery and resolved the issue just before seeing this.
The offending process seems to have been Freedom. I wound up using the brute force approach of greping for the more unique browser name strings (“Vivaldi”, “Sidekick”, “Microsoft Edge”) in every file under /Applications and ~/Library, and sure enough, the Freedom.app binary contains references all of them (and looking at where they occur they all seem to be used in the same way as an appName argument to something, which I presume is ultimately an AppleEvents call). The issue seems to have been resolved in a recent update to Freedom, so after forcing updates I haven’t seen the phantom alerts since.
Update: The Accessibility Inspector approach does seem to work! I just reproduced the headless-app-showing-a-selection-dialog-for-a-missing-app issue with a trivial AppleScript saved as an app (which I modified to LSUIElement = true) and the inspector was able to see the parent app of the phantom dialog window perfectly.