Updating fails (except it does indeed update)
My guess is that you're bumping into your php.ini file's memory_limit and/or max_execution_time setting.  It just happens to be towards the end of the update process where Craft is (for all practical purposes) updating, and it fails in the "cleaning up" portion of the update process.

Bump those up to something like 256M and 120 (respectively) and see if that helps going forward.


  Since the prior update pseudo-failure there was a message about maintenance mode and a button to continue (which I'm assuming is a flag set somewhere from the prior update). 


Yup, it's a flag set that never gets reset if there's a fatal PHP error (live above).  It'll go away after the next successful update.