The behavior works for me. Sometimes if I am setting the group tag in the portal, i need to change it a few times for it to update, then wait 15 min or so for it to update. After I change the group tag from the old group tag, it gets added to the other AAD group and then has the new appropriate Autopilot profile assigned. Have you tested it in PowerShell to see if it works better for you?
Eg:
Connect-MSGraph
Set-AutopilotDevice -id <ID> -groupTag <GroupTag>
Or even use PowerShell to see if it has set the new group tag or if it shows the old group tag just by using "Get-AutopilotDevice"