Set the table saw fence so the blade is centered on the seam, then push the whole thing through the saw. If the board edges are quite bowed or curvy, you may need a second pass. If so, unscrew the cleats, reclamp the boards across the middle center, then screw on the cleats again. The screws ding up the ends of the boards a little, so cut the boards you’re jointing a bit long, then trim 1/2 in. from each end to remove the screw holes.
Typically furniture such as tables and chairs is made using solid stock from hardwoods due to its strength and resistance to warping. Additionally, they also have a greater variety of grain patterns and color and take a finish better which allows the woodworker to exercise a great deal of artistic liberty. Hardwoods can be cut more cleanly and leave less residue on sawblades and other woodworking tools. Cabinet/fixture makers employ the use of plywood and other man made panel products. Some furniture, such as the Windsor chair involve green woodworking, shaping with wood while it contains its natural moisture prior to drying.