I do not agree that if the OC pays the bill it assumes liability. It has a statutory duty to repair common property which means pay for it either itself or through its insurer. That does not mean that it can’t take action to recoup those expenses.
I’m assuming the owners’ corp. has already paid the bill in which case it is a charge against the title of the unit and binds successors in title. Nothing to do with estate law and no need for a caveat on title.