Is is doable––Yes, but you will no doubt freeze and need to be removed from the mountain via helicopter . . . something like that . . . Actually, I would say that you could no doubt spend a few hours in shorts, during the afternoon, but you won't be doing any alpine starts in a pair of shorts.
http://www.snowcrest.net/camera/bigdaily.mpgShasta is not whitney. I've only done shasta three times, all in the spring and there was no way you'd want to be in shorts for more than a few afternoon hours.
I'm betting that even if there was very little snow on shasta it would still be pretty damn cold and windy.
Whitney, I've done twelve (end of JMT) times and slept up top maybe seven-eight times. It gets really cold during the night, but there's at least no snow and the conditions are not as apt to change drastically, like they can on Shasta. Go up thee unprepared and you're in trouble.