my big, big bugaboo is eating dinner as a family. I too have the crazy schedule - right now we have piano, drums, viola and competitive baseball (and this has slowed down because soccer and spanish are over.....and I only have 2 kids). but we also have always eaten fairly early, 5:30 to 5:45.
in years past, I'd do the dinner in the car or at the field thing but by the end of whatever sport season it was, it seemed as if we were all sniping at each other and behavior was not it's best (all the time, not just after the activity). and my husband was not connecting with whatever child he was not coaching and I was not connecting with the child he was coaching....
my solution this year (last fall and this spring) is fruit and an energy bar (or equivalent, such as a handful of nuts or peanut butter crackers or cheese and crackers) on the way to the activity. dinner in the oven (w/a timer) or a crock pot so that we can sit down together when we get home. many nights one of the adults is home and can make a "real" dinner. even if we only sit at the table for 20 minutes, everyone seems better off and better behaved over the long term. and, yes, we've tried the sitting at breakfast instead but it just doesn't work for us. primarily because my husband and I are NOT morning people.
yes, something has to give timewise. for us, it has been showers for children as often as I'd really like. and, pushing getting up for school such that there are mornings, one of us is literally walking the child through dressing, etc so we can get out the door in 15 minutes. but, we have been happier, kids have been better behaved and we have yet to have an illness this spring - meaning the kids are less stressed.
good luck.