(I am using XFCE in German, so maybe the wording I describe here is not 100% accurate)
In the XFCE settings menu -> Session management -> Automatically started applications, there is an Item "Policykit Authentication Agent".
This was unchecked. Re-check, reboot (probably logout and login is sufficient), and it is back working.
But: I am absolutely sure I did not remove the check. On one of the affected computers, I have never been in this dialog before.
I have found the solution by happenstance:
(I am using XFCE in German, so maybe the wording I describe here is not 100% accurate)
In the XFCE settings menu -> Session management -> Automatically started applications, there is an Item "Policykit Authentication Agent".
This was unchecked. Re-check, reboot (probably logout and login is sufficient), and it is back working.
But: I am absolutely sure I did not remove the check. On one of the affected computers, I have never been in this dialog before.