Hi there Sa6912,
If Screen Time isn't working as expected, you'll want to start by making sure your iPhone and your children's devices are updated to the latest software versions. For your iPhone, the most current version available is iOS 16.6. You can see the currently installed iOS version in Settings > General > About > Software Version. If needed, learn how to update your iPhone here: Update your iPhone or iPad - Apple Support
You mentioned that the screen doesn't seem to be responding on your children's devices when selecting options for Screen Time notifications; are they able to interact with the screen in other ways when this happens or does the device seem to be frozen entirely? If the device is completely frozen, try the steps in the relevant support articles below to see if they resolve the issue for you:
If your iPhone won't turn on or is frozen - Apple Support
If your iPad won't turn on or is frozen - Apple Support
If you need further assistance, reach out Apple Support directly. They're best-equipped to take a closer look at the issue and provide some additional guidance. This link will help you get started: Apple Support
Take care.