I dont know if compound bows are a Chinese invention; they are more likely from the steepes, and its around even in the middle east ancient world. The Egyptians acquired them from foreigners for example, before learning to make them on their own.
This is not to say that the compound bow is regarded as the most prized weapon in the ancient world. It takes 1.5 to 2 years to properly glue one together, and as the glue requires maturing. As such, compound bows are only made by the finest craftsmen. One Egyptian pharoah was said to take a personal interest on compound bows and would so often visit the craftsmen to examine the pieces.
The problem of compound bows is that they tend to get soft and fall apart in humid climates. Hence the English choose to use solid bows instead. The closer compound bow users get to the Meditereanean, the less effective the compound bow becomes. It probably has its effect as Atilla got closer to Rome, or during the Persian invasion of Greece.
I think in Japan, the Yumis were mainly solid as well. Humid climates in Japan and in Vietnam would have contributed to the lack of success the Mongol compound bow would have there.
The Chinese bow uses lacquer to help preserve the bow. Compared to other compound bows, it adds a new material to the equation---bamboo. Even though compound bows are widespread from central Russia to Korea, it is in the Far East, like in China or Korea where we see the big extreme 160lb plus draw strength bows. Most composite bows are small, intended for the horse rider. But when sedentary peoples use composite bows on foot soldiers, these bows tend to be big and powerful.
While compound recurve bows are from the north, the crossbow originated in the south. The crossbow is more of a southern chinese invention, and the concept may have originated from tribes south of the Yangtze. Unlike the compound bow, the bow on the crossbow tends to be solid, so it was easy to manufacture. What put it all together was the invention of the blast furnace, which allowed mass manufacture of the precise brass made trigger mechanism. Later these furnaces are instrumental in the widespread manufacture of metal stirrups.
The crossbow isn't a weapon made by artisans and craftsman, it was the first weapon in the world mass manufactured by a semi-industrial manufacturing infrastructure, the first military-industrial complex in the world. The result of this, you can pound out more crossbows than compound bows. And since it is a lot easier to train peasants to use crossbows than archers, you are producing a lot more soldiers. Professional archers like northern horseriders had a lifetime of proficiency to develop their skills, which is not available to sedentary agricultural peoples who main skills in life are to grow veggies. Chinese armies in general tend to use mercenary, allied or recruited northern horsemen as their archery component, but their crossbow troops are mainly from their main populations.
There is also a reason why China never developed a strong feudal structure. Feudal lords can afford their companies of blacksmiths to make weapons for them, and they can afford stable to grow horses. But they cannot afford the "factories" of blast furnaces---a monopoly by the imperial dynasty---that are needed to mass manufacture the trigger mechanisms for crossbows.
With the crossbow also came the end of the "romantic age" in China in swordmanship. it is because a simple peasant with a crossbow can kill mounted skilled warriors with their lifetime of training, their lavish armor and horses, their beautifully crafted swords, just with a simple press of the trigger. The crossbow is the great equalizer that lets the farmer kill the warrior and the knights. It meant the democratization of warfare and at the same time, the consolidation of power to those who control the industry.