I have looked and can't find any SIDs for this airport- I can find the STARs but no SIDs.
If there are SIDs, do any of them involve a right turn after takeoff, out across the lake?
The direct flight route would call for a left turn after take off, so unless he was following a SID which involved flight in the wrong direction initially (by no means unusual of course) it might be significant that his initial turn was the wrong way- i.e.- not positively intended. Then startle would play a part- he is expecting to see habitation and a biggish city, when all he sees is a black hole. METAR suggests IMC from 2600 ft, and last plot was circa 3000ft.
so fairly quickly into IMC, at a time of high workload, rapid acceleration (somatogravic illusion?) especially if his head was also turned, introducing a lateral component to the illusion=graveyard spiral.
All pure speculation but comments appreciated.
Thanks for the NZ CAA link vmandr - well worth a read for a GA single pilot