Yes, “Just” an MPU isn’t enough to let your whole program just ignore pieces that “crash”. It just lets you fail more gracefully in a safer manner.
If the task never trips the MPU, then the MPU wasn’t actually needed, which means for most programs, the MPU is really just a development phase support, perhaps kept in the application to avoid major changes at release, and to make safer the fail mode if something does go wrong. This can be important, but still, only affect things if something goes wrong.