After a little while, I realized that Catalina included some extra privacy protections. I went into System Preferences and turned on “Full Disk Access” for balenaEtcher. But lo and behold, the error still persisted.
After doing a bit of research, I found that balenaEtcher, and Etcher (the former name of balenaEtcher), had a command-line utility to launch the app.
The command booted up a graphical interface of balenaEtcher, the difference that, because of the “sudo” in the command, it ran under the root user. This bypassed the security/privacy restrictions placed (wisely) by Apple on Catalina.