This seemed promising enough that @verdre built a proper extension for it, which implements the exact visuals and behavior we wanted (though still very hacky code-wise).
This is what we showed off during the design state of the union talk at GUADEC:
We used this extension to extensively test the behavior, both on ourselves (by daily driving it) and as part of a series of informal user interviews.
One interesting learning from those interviews was that for a lot of people "abstract symbol I don't know" intuitively translates to "menu".
Even though the indicator doesn't have a direct reference to "apps", people tended to click the pill in the top left without thinking when asked to launch apps, since it looked like a menu to them.