Been off here a few days. Lots of snow so lots of hours plowing then sleep.
I dont think anyone has said that the web of a beam isnt important. OR that the flanges are more important than the web. Yes, they are all important and take away any part and you no longer have a beam.
But as far as strength is concerned, in the vertical direction only, adding material to the web does little to increase strength. But adding material to the flanges does add strength. Either wider, or thicker flanges increase the strength. Only other way to add strength to a beam is to make the flanges farther apart, IE: taller web.
Lets say you have an 8" beam with 4" flanges. And a spare piece of 8" flat steel. Cutting that 8" flat steel into two 4" pieces and adding them to the flanges would add more strength than adding it vertically to the web. Thats all I am saying.