Well, prolonged its demise a little.
My teenage son, despite repeated requests, frequently walks away from the computer leaving it running. And now that I've got a shiny new and very quiet computer it is not always obvious that it has been left running.
While formerly I would just changed the program the window manager invoked as a screensaver Linux systems have got a lot more convoluted. Whether this complexity is a good or necessary thing - I will reserve judgement.
Anyway, to cut a long story short, the solution for me was to use xautolock which, once started, monitors for a period of inactivity then runs a program. Setting it to start automatically on my Mint / mate box (no XClients / Xsession) was a matter of adding a .desktop entry to the XDG autostart directory:
Exec=xautolock -detectsleep -time 50 -locker "/sbin/shutdown -h now"