And Family Sharing, which is not new at all, was designed for this exact situation. You just haven't adopted it. It is the solution to a problem that became apparent several years ago in families. It is only going to get worse when Apple releases Group FaceTime. If it is "obvious" that millions of parents do this then I feel really worried for our future as we have a lot of parents that have no clue as to what they are doing. Maybe in this case you, and the millions of other parents, should let the kids take over and set everything properly.
Here are some alternate steps that may or may not work and could in fact cause more issues:
On each device go to Settings > Messages > Send & Receive At and tap the shared Apple ID to uncheck it. Do the same in FaceTime.
Turn off iMessage and just use SMS (green bubble) texting.
Contact Apple at Feedback - iPhone - Apple and ask them to completely revamp their Apple ID system to allow everyone to share Apple IDs with no issue on all devices.
This change has become apparent with this release most likely due to changes that were made for iCloud Messages, the upcoming Group FaceTime and security issues which Apple does not discuss. Since Apple long ago said you should not share Apple IDs they do not test new releases for that scenario resulting in unexpected results for the minority that do so.