Screentime on child's Apple Watch

We used Family Setup to set up my son's Apple Watch using my spouse's iPhone. My son also has an iPad associated with his Apple ID. Before he got the Apple Watch, we used Screen Time to limit the content he could access on the iPad. The same content restrictions are not carrying over to the Apple Watch -- he can use Siri to pull up videos or web results, for example. We have been unable to stop this.

In the Watch app, the Screen Time option just redirects us to Settings > Screen Time. (It says: "Screen Time is where you go to manage parental controls for all of your children's devices. Get insights about your family members' creen time and set limits for what you want to manage. Go to Screen Time Settings.")

In Screen Time settings, there are no Watch specific settings, the data that is reported is for the iPad, and the restrictions do not carry over to the Apple Watch.

Please help! I am usually pretty good with this stuff, but I am at a loss for what we are supposed to do. Thank you!