Toolbar's show() and hide() method do not properly toggle items visibility in the overflow popup.
The 'Button' is visible in both the Toolbar and the overflow popup initially.
When the Toolbar is resized so that the 'Dashboard' button is not visible, the 'Dashboard' button is also not visible in the overflow popup.
Buttons and button groups support badges only if they are "standing alone". If buttons or button groups are part of a toolbar item collection, badge system is not working/implemented. I don't see any reason why this feature would be missing from toolbar items unless it was maybe overlooked :) For the sake of consistency and usability I think toolbar items (at least buttons) should behave the same across.