I'd suggest a simple test: remix menu items and icons and test, if this has significant impact on usability. If not, the icons are just arbitrary decoration and ultimately add clutter.
Referring to the examples provided in the article, I'd suggest that the impact on the Safari app menu should be minimal (so these are non-functional icons), while the impact on the Move & Resize submenu would be devastating and should result in confusion (so these are essential).
If you can remix with minimal impact, don't do icons. (In the case of the app menu, these are apparently meant to add structure, which is already established by other means like menu separators, so you have now two – or, including indentation, three – systems of structure and visual hierarchy that are fighting each other.)
Moreover, if you put icons everywhere, you're forgoing the facility to convey state, like active state checkmarks, since these, instead of standing out and signalling change, would be just drowned in the decorative clutter. (So what's next? Add color and/or animation, like swirling checkmarks?) And this, BTW, is also why the icons in the Move & Resize menu are effective: they are conveying and illustrating state (in terms of a preview), while most of the other menu icons (mostly referring to activities) do not. So, as a rule of thumb: icons referring to state may be useful and even desirable, while icons referring to activities are probably better left out. (And, if you feel the need for something like bullet points to mark your most important menu items, there's probably a deeper problem with your menu structure.)
Referring to the examples provided in the article, I'd suggest that the impact on the Safari app menu should be minimal (so these are non-functional icons), while the impact on the Move & Resize submenu would be devastating and should result in confusion (so these are essential).
If you can remix with minimal impact, don't do icons. (In the case of the app menu, these are apparently meant to add structure, which is already established by other means like menu separators, so you have now two – or, including indentation, three – systems of structure and visual hierarchy that are fighting each other.)
Moreover, if you put icons everywhere, you're forgoing the facility to convey state, like active state checkmarks, since these, instead of standing out and signalling change, would be just drowned in the decorative clutter. (So what's next? Add color and/or animation, like swirling checkmarks?) And this, BTW, is also why the icons in the Move & Resize menu are effective: they are conveying and illustrating state (in terms of a preview), while most of the other menu icons (mostly referring to activities) do not. So, as a rule of thumb: icons referring to state may be useful and even desirable, while icons referring to activities are probably better left out. (And, if you feel the need for something like bullet points to mark your most important menu items, there's probably a deeper problem with your menu structure.)