[Moving the discussion from email]
So, the icon clearly is not updated for a working timer in Windows 10 21H1 for whatever reason and if this is so for a couple of Windows 10 releases, it's unlikely to be fixed any time soon. Also, the existing icon is not informative - the round arrows for a working icon (even if they were displayed) are not easily recognizable from the first glance (you need to consciously look at the icon and pay attention to what is displayed).
My suggestion is to leave the icon without updating (if updates don't work anyway) and instead, change the caption of the taskbar entry as follows:
For running tasks:
{Task name} – WorkingTime
For paused tasks:
Paused – {Task name} – WorkingTime
And for stopped state –
Stopped – WorkingTime
This will let us clearly see the task name if it is active or see that the timer is paused/stopped when it is.