I don't think that the aircraft ran out of fuel, the preliminary report here:
http://www.bfu.admin.ch/common/pdf/HB-GDS.pdf
states that it had only been airborne for less than 30 minutes and that fuel leaked from the wreck.
This is a rumour network so I will speculate:
I know that the weather at the time was benign, clear skies and light winds. I would assume that the father was seated forward with his wife and children behind. He twisted round in his seat to speak to his family behind him and in doing so inadvertently pressed full rudder. This set off a chain of events leading to loss of control of the aircraft ending in a flat spin, perhaps even with structural damage before impact. Unlikely? No, it has happened before.
The aircraft first flew in 1962, so would have been later fitted with GPS. I am sure that the investigators will be trying to read the memory, if any, of this and certainly of any mobile phones or cameras that often include a GPS function in order to get an idea of the flight profile.