Nice chair... My FIL made a couple of chairs very similar to that and a matching table for our kids.
They lasted for years... but in the later years, the plywood legs got weaker on the bottom and in the middle vertically... May I suggest treating the ply to make it stronger like epoxy that would soak into the wood to really stiffen it...
I have had to repair them by cutting other slim pieces to replace missing sections of the ply... Good and strong now... Kids are grown & out... used by grandkids now.
That drawer idea sounds good!!