As discussed in T12778, the icon can never be in a correct state: either it's
redundant with the icon you're hovering over that made the tooltip appear, or
else it's different from the icon you're hovering over, which is inconsistent.
Furthermore, some instances of the tooltip have icons and others do not,
heightening the inconsistency.

Since either case is undesirable, let's just remove the icon from the tooltip
entirely. This reduces the visual noise in the tooltip and solves the entire
class of bugs automatically.