A bass is the easiest thing to bring a backup of and remains fully in your control. Any other failure, amp/cab/pedals etc, it's more than plausible that the sound tech can cover you for that, even if it's just straight DI to FoH, depending on the type of gig. If it's a gig where you are running everything, then yes a backup line needs to be drawn somewhere, but ultimately if something in your control fails, I assume you would not charge for the gig.
Re the PA example, obviously any equipment that isn't yours is beyond your control and responsibility, so doesn't really enter into it.