How old were your children when they figured out they could control Siri? One of my kids tried it once but Siri couldn't recognize the voice and refused to respond.
Other possible "solutions" I read about that aren't as good as what you're currently doing:
- Keep Hey Siri on, but create custom shortcuts to automatically respond with your preferred admonishment for things you don't want them asking over and over again
- Keep Hey Siri on, but set HomePod volume to 0
(By the way, thanks for mentioning HN Notify in your profile. Had no idea that existed. Now I'm using it.)