I recently made table and two stools for my granddaughters aged 6 and 8. Table frame was made with top grade Scandinavian pine. Haunched mortise and tenon joints to legs for top rails with mortised and tenon lower rails to give added stability. Top was screwed to base through wooden braces you mention. legs were 11/2 ins square and with M&T joins will take anything the kids will throw at it. I made stools from same pine per Paul’s bench stool project, scaled down ,but with circular seats.
Re size the table measures 600mm by 1200mm and kids sit at each end giving them their own work area.of 600×600. They love it .
Sizing was bit of guess work. I went for a very slightly lower than a standard dining table height which the children can sit at fine albeit the younger one has a cushion. The end result is it works for the kids and adults can sit comfortably at it as well.
Hope this might help. Pic attached.
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