Interesting. I opened a service ticket with apple yesterday and talked to an Apple Support Agent. I was told the screen time settings are always stored in the Family Organizer Roles iCloud Account.
In my case, thats my wife. So I deactivated everything screen time with her account on her device for the kids and configured everything from scratch. Seems to work, but after a couple of minutes, settings are lost again.
So it seems like, if its true what the Apple Agent said, that there is a problem to store the settings permanentely in the family organizer screen time settings and therefore the config is not available on the kids devices.
Will try the one minute more option though. Thanks for the tip!