Hi @Heiko Nolte ,
I have tried to reproduce the issue with the same good script that you have shared but I was successfully able to see the tag being updated in the resources of a resource group. In my case, I even didn't had to wait for long for the change to reflect; it was instant.
If the issue still persists for you, please confirm if you have received any error while executing the script. If No, then as a basic troubleshooting, write-host the variables at regular steps and see if the variables hold the required values or not. Or try out for another test resource group with one or two test resources to see if the prior issue is with one particular resource group or not.