I think I've solved it. It turns out that Titanium Backup (free edition), closes apps before backing them up, whereas the Pro edition can just "pause" them. I just ran a backup and it seems that Automate is still running.
However, I should say that most other apps (such as Whatsapp, or Withings Healthmate) seem to be able to carry on working after being killed in the same way:
03-29 20:05:02.309 I/ActivityManager( 939): Force stopping com.whatsapp appid=10284 user=0: from pid 22841
03-29 20:05:02.310 I/ActivityManager( 939): Killing 10556:com.whatsapp/u0a284 (adj 7): stop com.whatsapp
03-29 20:05:02.415 I/WindowState( 939): WIN DEATH: Window{ed8eb40 u0 com.whatsapp/com.whatsapp.HomeActivity}
03-29 20:05:02.524 W/ActivityManager( 939): Scheduling restart of crashed service com.whatsapp/.messaging.MessageService in 1000ms
03-29 20:05:02.530 I/ActivityManager( 939): Force finishing activity ActivityRecord{29d6cb3c u0 com.whatsapp/.HomeActivity t21905}
03-29 20:05:02.537 I/ActivityManager( 939): Force stopping service ServiceRecord{248782e3 u0 com.whatsapp/.messaging.MessageService}
03-29 20:05:02.542 W/ActivityManager( 939): Spurious death for ProcessRecord{15b3731b 10556:com.whatsapp/u0a284}, curProc for 10556: null
Healthmate (which counts steps, amongst other things) was killed but then seemed to restart and was able to count my steps after the backup.