MacOS – Why doesn’t the screen saver delay that I set persist after restarting the Mac

I cannot tell about screensaver for sure, but you can use pmset -g to show which process is disturbing sleep (think this first appeared in 10.6.7)

# pmset -g
[cut]
sleep       0 (imposed by 1517)
[cut]

this means that pid 1517 is preventing the system from sleeping.

# ps ax |grep 1517
1517   ??  Rs     1:59.76 /System/Library/CoreServices/backupd 

This shows that this is the backup daemon running.

If your computer is locked, the screensaver settings for the root user account is used instead of your own. You can change those settings according to the answer to this question: How can I set the Mac OS X screensaver when not logged in?