If the auxilliary drive shaft fails, does that mean that bits of the drive shaft are flailing about and maybe causing serious damage to nearby components?
The "serious" damage that is covered in the training relates to issues directly affecting safety of flight and are not related to economic issues or cascading effects internal to the engine.
Experience has shown
In a massive engine failure like disk rupture, case rupture, turbine shaft fracture, the serious safety related damage is all over by the time your eyes glance at the engine gages. The concern at this point is fire and that's about all the control that you have on it from the cockpit. even shutdown it will continue to windmill and possibly set up felt vibrations, but this is not of concern structurally to the aircraft since the design of the aircraft can accomodate this.
Many engine failures (including flame outs) can be restarted and at least operated at idle (generators, hydraulics). However, I would hestitate to run one that manifests vibration associated with other symptoms of failure.
It's easy to be cautious about an engine failure if you still have aircraft performance margin, but if you're in a tight situation you probably can continue a sick engine at idle (if no vibration) without the worry of catastrophic failure.