The other half of this is that your real voice — your clean singing voice — needs to stay in the picture.
The clean voice is what gives the scream its note and its power. It's the foundation everything sits on. When you take the clean voice out and just push air, you lose the pitch, you lose the body, and the whole thing collapses into noise.
What you're aiming for is a scream that has both: the tonal foundation from the vocal folds doing their job, and the distortion layer from the false vocal folds and arytenoids being engaged on top of it. The clean voice underneath is what makes the scream actually sound heavy and full instead of thin and airy.
Try this: start screaming a note, and without stopping, gradually let more of your clean singing voice come through. The scream should get bigger and more solid, not cleaner. If you can hear the tone improve and the distortion hold at the same time, you're in the right place.