Clicking on the chip and not having the window come to the foreground is a fairly weird issue. Do you have anything else installed that uses Accessibility permissions?

On the other observation, it works this way because apps on macOS do not usually quit when you close all of their windows. If you start an already open app again when it has no windows open, it will bring the app into focus, and you will see it in the menu bar.

Regarding your recommendation, I think I’ll need to experiment with it a bit, since it’ll be important to differentiate between pinned apps and apps with no windows in the bar.