I turned off LogMeIn - problem remains. So, I shut down and booted into "Safe Mode", twiddled my thumbs for quite a while, then rebooted normally. After the timer expired - Lo! - the screensaver started up! And the weird console messages stopped.
I re-enabled LogMeIn. Waiting ... All OK.
Starting AA ... All OK!
Moral of the story, boys and girls: Safe Mode is our friend.