There is not another option. A child that is intent on disobeying your clear instructions can always find a way if that is their desire. For example if you block one contact, they can call someone else who can initiate a 3 way call for communication which is undetected by the Childs device. If you blocked FaceTime for sharing the screen, there are also third party apps that do the same kind of thing such as Zoom that they could use. Sharing may give your child live view, but there would still be nothing stopping the other person from recording the content on Social Media and sending it to your child as a Video.
Screen Time is just a tool it is not the end all for managing a child's activities. The child should see that having the iPhone as a privilege that can be taken away if they are not respectful to your rules. You can still get the "dumb" phones for about $20 that don't have apps/internet access and used solely for communication, and given that option you may find a child to be much more compliant.