Please correct me if my assumption is incorrect, but it seems the Chinese (or other Eastern cultures) never figured out plumbing until the 20th century, despite being quite advanced throughout ancient times in many other forms of technology. In contrast, the Greeks and Romans had sewage systems millennia ago. Was this actually the case? And if so, why?
The Chinese had a very well developed and practical system for management of human waste.
For a first-hand observer's description how it worked, written by an American agricultural scientist in the early 20th century, I refer you to chapter 9 of FH King's classic 1911 work Farmers of Forty Centuries; Or, Permanent Agriculture in China, Korea and Japan.
In brief, human waste was meticulously collected and utilized as fertilizer, enabling sustainable high intensity agriculture in an era before chemical fertilizers:
"In the Far East, for more than thirty centuries, these enormous wastes have been religiously saved and today the four hundred million of adult population send back to their fields annually 150,000 tons of phosphorus; 376,000 tons of potassium, and 1,158,000 tons of nitrogen comprised in a gross weight exceeding 182 million tons, gathered from every home, from the country villages and from the great cities like Hankow-Wuchang-Hanyang with its 1,770,000 people swarming on a land area delimited by a radius of four miles."
Building sewage systems to gather human waste and flush it out to rivers or the sea would have been considered insanely wasteful.
I should note that this method of waste disposal did come at a price: parasite infections and epidemics of dysentery and cholera. In a 1994 survey of Chinese parasite disease prevalence a majority of the population was found to be carrying one or more parasites, with infection rates over 80% in some southern provinces.