You have to think about the extent to which you will be able to accept any bad decisions that your children make within their budget. If a child decides not to buy a coat or new shoes, can you live with them being cold or wearing ragged old shoes? If they haven't saved enough for school supplies towards the end of the year, what will you do?
I couldn't live with this plan. There are certain necessities that it is my responsibility to purchase for my child and that includes basic, weather appropriate clothing and school supplies. If my child wasn't able to or didn't want to buy those things, I'd still have to buy them. I think helping your child keep track of what was spent is a good idea. I would not let my child make financial choices if they couldn't live with the consequences because I am ultimately responsible for the consequences. That is the difference between a child and an adult.