I don't think that Damon can shorten orthodontic treatment time (the duration you spend wearing braces) significantly because it does not change the fundamental biology of how teeth move.
Orthodontic tooth movement is a cellular process of bone remodelling in response to forces on the tooth. The force can come from a brace (like Damon or Invisalign) or from a finger (like when a child sucks on their thumb). The force needs to be present for more than 8 hours in order to trigger bone remodelling.
At the level of the tooth (where the braces is attached), only force direction and force magnitude (how strong the force is) matters. Remember that the tooth has no idea what type of appliance (Damon, conventional braces, Invisalign etc) is moving it!
The main difference between Damon and conventional brackets is how the archwire is attached to the tooth. Damon has a self-locking system while conventional braces use rubber elastics or metal ties. In the end, most of the time-saving come from how quickly the archwires can be changed (how much time you spend in the dental chair) instead of how quickly the teeth can move through the bone.
Osteoperforation through periodontally-accelerated osteogenic orthodontics (PAOO) or Propel, makes use of inflammation to accelerate tooth movement. Creating holes in the bone makes moving teeth easier since bone becomes soft during the healing process. This is a biologically-plausible mechanism to move teeth more easily and quickly which has been objectively proven in clinical studies.
