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Step-by-step build — 5 steps. Also available as a print sheet.
Cut the plywood scrap to an 8-1/2" x 8-1/2" square, then clip all four corners off at 45°. This plate is what the seat's flange mounts to.
Set a circular saw blade to a 3/4" depth and notch two of the footrest pieces along their long face, and the other two along their short face, so each notch is half the material's thickness. Glue — don't screw or nail — the pairs together where the notches interlock; a fastener here would split the thin remaining wood.
Assemble the legs, top stretchers, and mounting plate into the stool base using 1-1/2" pocket holes and 2-1/2" pocket hole screws, or countersunk 2-1/2" wood screws if you'd rather not drill pocket holes.
Attach the glued footrest cross-support assembly to the outside of the legs with a single countersunk 2-1/2" screw per joint plus glue — one screw is enough here since the half-lap joint is already doing the structural work.
Mount the base flange to the top mounting plate on the stool base, and the matching flange or top plate to the underside of the round seat. Thread the all-thread rod into the base flange, secured with a nut and washer, then thread the seat onto the rod. Add a second nut and washer above the seat's flange to act as a stop and lock the height once you've dialed it in.