Lightroom 6.2.1 is still crashing on exit (Mac) regardless of whether you skip the back-up or not. If backed up, the screen disappears as expected, but Lightroom is still 'live'. E.G. If you try and run it nothing happens because it is still running or in fact not responding.
If you exit by skipping the back-up, the beach ball spins for ever and a forced shutdown indicates that Lightroom is not responding. Come on Adobe, sharpen up your act!!
i am not getting this behavior myself on OS X El capitan. What Os are you on and did you reboot your machine in between? This sort of behavior is sometimes caused by parts of the program being kept around by OS X in cache. A reboot will help in that case.
An Adobe employee said that LR 6.2.1 may have fixed this issue, but at least one other person has reported that it didn't resolve it for him. So please add your vote and details to this bug report in the official Adobe feedback forum: Lightroom 2015.2 hangs on quit on MacOS 10.11. That will make it more likely Adobe will prioritize the fix. (This forum here is primarily a user-to-user forum.)