Arduino Due not executing code after power on
After some more research I found out that

a) the issue applies to some Due boards only, not all

b) newer boards seem to have fixed the issue

c) there's a software (firmware update) and hardware (10k resistor) solution

I went with the hardware fix shown here:
http://www.dimdim.gr/2016/03/fixing-startup-issues-with-arduino-dues/

I did not have an SMD resistor at hand, so mine looks ancient but does the job.
Hope that helps others, too.



addendum:

The problem occurs only on some DUE boards and is due to some undocumented behaviour of the ATSAM3X8E processor combined with the behaviour of some MOSFETS installed on the DUE boards. So its occurence is largely a matter of luck.

The fix is simple: you just solder a 10K resistor across the top of this mosfet:

resistor about to be soldered on



resistor designated as R99