Why couldn't Air Moorea Flight 1121 use pitch trim to land safely?
All quotes in this answer are from the English translation of the BEA accident report.

The elevator cable broke at "an altitude estimated at between three and four hundred feet",(p. 7) and there were eleven seconds between that time and the point of impact. If it had happened higher, the pilot would potentially have had time to figure out what happened and pull out of the dive with trim. Taken by surprise and not knowing what happened, the pilot simply did not have time to recover.

The Twin Otter would have been demonstrated to be recoverable from an elevator cable break as part of its certification (p. 17):


  The CAR 3 regulation, basis of certification for the DHC6, requires a flight test demonstrating the aeroplane’s capacity to land using only the elevator trim in the case of failure of the primary longitudinal control of the aeroplane.


However, there's a big difference between "can be landed using just trim" and "can be recovered by a pilot who is surprised by the emergency". Flight tests performed during the investigation showed (p. 63, emphasis mine):


  At [the height the incident aircraft was at], only immediate action on the trim located on the centre pedestal would make it possible to recover the airplane. The test also showed that from level flight it takes about three seconds for a pilot trained for and prepared for this exercise to recover the airplane. 


Unfortunately, on this flight there simply was not enough time to discover the problem, figure out what went wrong, and recover (p. 63):


  The highly dynamic nature of the events following the failure must be emphasized here. Eleven seconds passed between the pilot’s exclamation and the impact. This only left a short time for the pilot to analyse the situation and apply a solution that he had to improvise. In addition, the stress associated with the airplane’s attitude and the difficulty in estimating his height, in the conditions on the day, in relation to the surface of the water, certainly affected his powers of analysis. The pilot was not trained or prepared, either during his training or during type rating, as indeed most pilots aren’t, to react to a loss of pitch control. Only a reflex action could thus have
  allowed him to recover the airplane before the impact.




To answer the edits to the question, note that the airplane was not trimmed to "fall out of the sky". The pilot called out the trim checklist item before flight(p. 17), and so should have set it to the takeoff setting. Presumably this would require gentle back-pressure for rotation and climb. The event that initiated the failure process was retracting the flaps(p. 39). This changed the flight dynamics, and therefore the elevator control loads. The flight testing discovered that this created a "pitch-down moment"(p. 39):


  To counter this effect and maintain the aeroplane on its initial trajectory, it was necessary to exert considerable pitch-up effort on the elevator control.


This is not a cause for concern. The procedure is to hold pressure on the controls and adjust trim to relieve the pressure. However, it was at this point that the elevator cable snapped, so the pilot was interrupted and didn't complete the process. If the incident had happened at a higher altitude, the airplane would have recovered itself from the upset (p. 42):


  [S]tick free, the flap retraction led to a pitch-down moment varying the aeroplane
  trim between 20° and 30°, the rate-of-climb indicator going to stop at less than
  3,000 ft/min and speed reaching 140 kt in twenty seconds, then the aeroplane
  recovered on its own in level flight in five seconds, the total loss in height being
  seven hundred feet[.]


To the last question, flying inverted would not be necessary for recovery. I do not know the maneuvering characteristics of the DH-6, but I am unsure that an outside loop could successfully be completed starting at 300-400 AGL.