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Step-by-step build — 6 steps. Also available as a print sheet.
Screw each leg cap across the tops of two legs with the 2-1/2" screws, making two identical U-frames. These are the box's whole skeleton — check them for square, because every slat that follows registers off them.
Attach four side slats to each frame with the 1-1/4" screws, leaving a 3/4" overhang at the top — the plan's trick is to use the 1x3 lid-support stock as a spacer gauge for that overhang. Keep the slat gaps even; they're doing the ventilation work.
Stand the two slatted sides up and screw the front and back slats across them — measure the actual box for these, cutting to fit rather than trusting the list. If you want the notch for the grill's hose at the front, mark and cut it now.
Lay the 1x3 supports on top of the leg caps, centered, and set the two outer lid slats on them flush with the box's outside edges. Screw the slats to the 1x3s — not to the box — with the 1-1/4" screws. You're building the lid in place so it fits this box, not a theoretical one.
Set the remaining lid slat in the middle, gap it evenly, and screw it to the supports. Lift the finished lid off and sand it well — this is the table surface, the part hands and drinks actually touch.
Attach the lid to the box with the piano hinge (or a pair of exterior hinges) along the back edge. Swap your tank in, check the lid opens past vertical so it stays put while you work the valve, and park the table next to the grill.
Leave the cedar raw to silver, or oil it — but never seal the box up tight. The gaps are the safety feature.