I then looked through all open issues in Electrum GitHub, and found this:
"Silent crash on Windows7 if path to AppData\Roaming\Electrum contains national characters" (https://github.com/spesmilo/electrum/issues/2376)
It appears the cause may be that my user name contains a national character, i.e. the Danish letter "æ" (a+e ligature).
Apparently this causes Electrum to crash silently. I made a comment on the issue - hopefully the bug will be fixed.
Meanwhile, I can run the portable version of Electrum, since this will allow me to start in any folder, not just my home folder.
Thanks for your help.
Interesting bug, but that definitely seems like it is the issue. It would be nice if it was possible to start Electrum in a debug mode or similar that allows outputting errors/crashes to a log file which could be useful for diagnosing issues like this when they appear.