Minor mistake: the Start menu is center justified, with the actual Start menu icon on the left. The widget should appear on the right, just to the left of the tray.
Moreover, the problem is somewhat intermittent. Most of the time the widget menu doesn't appear, but sometimes it does on reboot.
Apart from the workaround described above (Explorer restart), I haven't been able to fix the issue.
Toggling "Enhance the taskbar" re-enables the widget, but then doesn't always keep the widget when toggling back on. Perhaps just toggling back and forth can reproduce it, as it has for me.
Nothing else to my knowledge is configured to edit the taskbar.
Win version: 24h2 26100.6725
Start11 version: 2.55 business edition