The amount and type of data sent back from the aircraft is determined by the customer (the airline) options selected; i.e kit fitted and service level agreements with Inmarsat, etc.
If I recall correctly, MAS had opted for only very basic data reporting services through Inmarsat, so once the aircraft was out of range of land based VHF ACARS sites options for transmission of data was limited. Add to this that the ACARS system on board appears to have been disabled, either through deliberate interference or accident, then the lack of data from the aircraft isn't surprising.
Some insist that the fact that ACARS was disabled, but not the SATCOM radio points to deliberate interference; it's possible to just log off ACARS from the flight deck, but disabling SATCOM would have involved a trip to the aircraft's avionics bay (located below deck and accessed via a hatch outside of the flight deck) to pull circuit breakers. It's been suggested that any fire would have taken down all systems equally, including (as you pointed out) the autopilot, causing the aircraft to crash a lot sooner than it did.
It's a weird case and I don't think we'll ever know the full story, even if the flight recorders are recovered. The CVR has a loop time of 3 hours, so any voice recordings of the beginning of the incident has likely been lost. The investigation will have to rely upon the FDR, that has a much longer recording period.