You missed the most important disadvantage of a mechanical compressor and that is the increase in specific fuel consumption due to the not-insignificant amount of crank power needed to drive it.
As for performance being the only goal as a justification, a supercharger delivers a fixed pressure ratio and airflow depending on the engine RPM, VE characteristic and the drive ratio (pulley size); it is as little or as much power as the chosen charger and pulley, a bit like the choice of turbo.
People are pushing TDIs to 400 HP turbo only, and if low-end torque and minimum lag are of primary importance, turbos can be sized to suit that too, or a 2-stage sequential system to have your cake and eat it too.
Putting a blower in a TDI is a novelty and curiosity but doesn't put us into any previously uncharted performance territory not already achieved with turbos, but I'm honestly looking forward to someone stepping up to doing a very well thought-up system using a 3.0 TSFI unit (latest and greatest Eaton TVS series). This could possibly be done relatively straightforwardly in a PD since it doesn't have a separate pump pulley, but using a longer timing belt from an ALH or CR, and designing and fabricating a mounting bracket for the supercharger and any necessary idlers.