SMS/MMS requires a paired iPhone, as the Apple Watch cellular radio is LTE "Data" only, which means it does not have the standard cellular features that SMS uses.
See:
Use your Apple Watch without your iPhone nearby - Apple Support
2) To receive SMS, MMS, or push notifications from third-party apps on your cellular Apple Watch, your paired iPhone must be powered on and connected to Wi-Fi or cellular, but it doesn't need to be nearby. You also need to be signed in to iMessage on your iPhone to use Messages on your watch.
Your child does not have a paired iPhone, so just Internet based messaging services.