Don't get me wrong, I'm all for privacy. But between setting up the birthdate when creating my children's local account on their computers, and having to send a copy of their ID to every platform under the sun, I'd easily chose the former.
I'd even agree to a simple protocol (HTTP X-Over-18 / X-Over-21 headers?) to that.
I don't think there would be any difficulty with a kid setting up a computer, as in most juristictions the parents are responsible for their childrens' actions until they are adults themselves. So the oarents would still be responsible for what the kid did with the computer in the same way they often are now.