Either a particular system is fixed, or you have a very few short weeks to implement a backup plan. Perhaps run alternative or competitive applications witht their data sets in the background as a failsafe. Not too difficult, if focused on constraining the scope of the problem. It s the core application for figuring out who s supposed to be where, when. This isn't really a problem.
Computers break all the time.
They can just fix it when it fails. No big deal, everything is fine. I think risk still is the issue. They're just betting that significant errors won't pop up, or if errors do pop up, they will be easily worked around and fixed. That might be a foolish bet, but it's one that they seem to feel increasingly comfortable with.