When the "List All Tabs" button is on the right, it seems that TabKit automatically hides it. I right-clicked on the bar and clicked "Customize" and dragged the button to the left of the open tabs. Now when I exit customize mode, the button is still there. When I click it, it displays a vertical menu of all open tabs.
However, the list of tabs in the menu is not color-coded according to the grouping done by TabKit. So only that remains to be worked on.