It wouldn't collapse [all of] the financial industry. Any banks which may already be insolvent without FRB loans would have to declare bankruptcy. There are probably a couple solvent banks out there. Failure seems to be a fair and natural outcome for businesses that are poorly managed or choose unsustainable business models.
The FRB can remove the slosh by not rolling over loans to primaries when they come due, and by putting back the trash securities they took as collateral through the TAF and TSLF.