I see a few possibilities.
First, and most obvious, as mentioned in the previous answer, being in a body seems to make the Mind Flayer more vulnerable. For instance, not only did the Demodogs die in the previous season without being reactivated when the gate reopened slightly, but if we consider that the Mind Flayer likely controls them the same way as it controls human beings, through putting pieces of itself in them, as evidenced by their instinct-driven behavior before coming into contact with it, then we see that the Demodogs don't seem to have released pieces of the Mind Flayer when the gate closed. This kind of makes sense: if the Mind Flayer has incorporated itself so strongly into the brain (or nervous system, or soul, or whatever, considering the...pieces) of a creature that the creature dies when the link to the main Flayer is cut off, it's logical that the backlash would kill the Flayer piece. Note also that the underground tentacle extensions of the Mind Flayer seem to have died permanently when the first gate was closed as well.
As a further piece of evidence in this regard: no Mind Flayer dust left Billy's body after the main Flayer body killed him, even though he was still possessed by the Flayer, albeit temporarily in control of himself. That portion seems to have died with him.
Another possibility is that perhaps the Mind Flayer piece was still left in the body in an inert state, but that the US government didn't take any chances, and destroyed the body of the Flayer-monster, presumably with fire, the same method they used to kill the aboveground intrusions of the Mind Flayer. The Mind Flayer probably wouldn't be so frightened of heat if it couldn't do it actual harm. Of course, it's hard to do this if the Mind Flayer is moving around, trying to kill things, but once it was inert they could give it a try.
It's also worth noting the possibility that there's a minimum "size" of Mind Flayer piece that can survive the trauma of closing the gate. Presumably the piece in Will was rather large, giving the Mind Flayer full access to his mind, whereas the other pieces may have been smaller. Although not directly relevant to the situation in Season 3, it's possible that dividing itself up between Billy and the main Flayer body was bad for its health. It's also likely that the more time a piece spends disconnected from the main controlling mind (from the gate being closed), or perhaps the more times it's disconnected from said closure, the worse the state its going to be in.
Finally, keep in mind that even if they are not sure that closing the partly-open gate will kill that Mind Flayer piece for good, it's the best option they have. The only alternative is to leave the Mind Flayer free to wreak havoc and possibly acquire new victims, which is obviously no choice at all. They can close the gate immediately and later worry about what to do.
Perhaps the only way to stop the Mind Flayer for good is to destroy the main entity, in the place where it resides....