In and Out of the Band
The short answer is, 'now that these assholes are gone, everyone will do what I tell them.'
There is no "Nine Inch Nails." There's Trent Reznor, and then there are the people he hires to play with him. There is no significant difference between him and Billy Corgan. They're just trying to find people who aren't assholes who will help them make music.
This is not a bad thing, but let's dispense with the idea that these are bands. At this stage of the game, being over twenty years on for both artists, there is no point in hiding behind the facade that these are actual bands where people have to agree on the way forward or compromise with one another. If these were bands, the departure of two members would mean that the band cannot continue under the same name. In the case of the Smashing Pumpkins, Billy has either kicked out or driven out everyone and it is no longer a band. It's a brand name that helps him sell more of whatever he's selling.
To continue under a different name may seem artistically viable but it is, in reality, a difficult proposition for fans and artists alike. No matter how savvy people claim to be, they still don't know that Roger Waters was in Pink Floyd, and that if they go see Roger play live, he will play a lot of Pink Floyd songs.
Reznor and Corgan should just go out under their own name and their fans will know what they're going to get--Trent and Billy songs, played with backup players.