The unified banner's mode selector wasn't working on mobile, the menu
wouldn't open on touch. This is because the mode menu didn't specifically
handle touch events. We now handle touch events and make sure the menu
works as normal on mobile.

Testing Done:
- Opened and closed the menu on mobile, tapping on different parts of it
  (the label, icon, empty space).
- Switched between menu modes by clicking on the menu items.
- Made sure the menu still worked on desktop.

Reviewed at https://reviews.reviewboard.org/r/14129/