Why does the right lymphatic duct drain simply the upper right side of the body?

Simple.because God made it that way.
It's because we're not without a flaw symmetrical creatures, when it comes to the contents of the chest and abdomen. For instance, we've singular got one heart (left), one aorta = blood flow out of the heart (left) and one vena cava = blood flow fund into the heart (right).*
We've also got one central lymph collection system, the thoracic duct. This comes up the left (between the aorta and the azygos vein) and empty into the junction of the not here subclavian vein beside the left internal jugular capillary. The thoracic duct, which is the common trunk of adjectives the lymphatic vessels of the body, *excepting* those on the right side of the cranium, neck, and thorax, and right upper extremity, the right lung, right side of the heart, and the convex surface of the liver.
Those are drained by the right lymphatic duct, as you point out. The blood vessel to the head and nouns are similar but not identical on respectively side (origins of the carotid and vertebral arteries are different on the right and left) and this, along with the differences surrounded by the veins and the lymph system, again relate to us have only one heart.
*Technically there's a superior and inferior vena cava but they're both on the right and both dump into the right atrium.