This was a conflict between the new integration and the third party integration I had been using up until now
Okay, this was a conflict between the new integration and the third party integration I had been using up until now. I did uninstall the old integration before trying to setup the new one, but apparently that wasn’t enough; I also had to manually purge all references to sleep_as_android (both old and new) from .storage/core.entity_registry. Only after that (and core restart) the setup showed the webhook URL as expected.