Are you sure? While we all know PCs routers etc have log files which almost always log a shutdown request, these are only visible if you have access to the PC (or wreckage in this case). I am pretty certain there is nothing detectable outside the aircraft that lets you know if the RT, transponding or ACARS was powered down at the device or lost power. I don't have any specific details on the Satcom, but all of the information so far suggest the link communications are a logon/authenticate and then periodic status checks, but no expectation of a logoff or shutdown message when the system is powered down normally. So that to gives no clue as to why it stopped responding.
I would certainly expect the FDR to be able to answer many of these questions.
In any event, it is moot as I don't think there is any sensible scenario where someone is carfully following a power down protocol.