The showerhead aerates the water and spreads it around a large surface area , the other faucets don't spread the water around, they keep it in a stream so there's less likelihood that you'll smell it anywhere but the shower.
Let's face it, the only possible source of the smell is the hot water heater (or you would smell it in the cold water in the shower. The inner surface of the HWH tank is contaminated. If you read your owner's manual, it will tell you to do exactly what I suggested. However, if you know more about the HWH than the manufacturer, maybe keep trying "your way"...... Up to you, but I'd follow the manufacturer's advice. Double check your owner's manual, you might find some interesting solutions to your problem....
