They will start getting experience with arbitrary, petty, random and chaotic behavior as soon as you start sending them to public school.
It's great training, and not to be discounted. But having a haven of stability in the home is an essential counterbalance.
Based on my own childhood, I both totally agree with the premise and come to a fairly opposite conclusion: home must be a haven since it is the only place that has any reasonable chance of being a haven. School almost certainly won't be; other kids can be real assholes sometimes.
I've seen a lot of people that went to private schools really not understand just how stupid can be nor how such people think, because their world was chopped off at something like a 105 IQ floor.