Hello hartsja,
Welcome to Apple Support Communities. You should not be concerned with iCloud Private Relay feature being enabled on your child's device. It actually prevents websites from building a profile of their location and browsing history over time. There is no contradiction between this feature and parental controls or using Screen Time.
When you subscribe to iCloud+, you can use iCloud Private Relay to help prevent websites and network providers from creating a detailed profile about you. When iCloud Private Relay is on, the traffic leaving your iPhone is encrypted and sent through two separate internet relays. This prevents websites from seeing your IP address and exact location while preventing network providers from collecting your browsing activity in Safari.
Protect your web browsing with iCloud Private Relay on iPad - Apple Support
For more information, please check this out:
About iCloud Private Relay - Apple Support
We hope this information is useful. Kind regards.