I just had an enlightenment:

The reason why it's so valuable to me to sleep until I wake up naturally is because it calms me down. When I wake up naturally after a complete sleep, my mind is blank and I'm not worrying about anything that does not need to be addressed immediately. When I have not had enough sleep, I find myself fretting about things that are beyond my immediate control and/or inevitable. So the reason I often wake up mornings craving more sleep is not so much because I'm tired, but because I want to be calmed.