I watched a documentary called Chinese Chariots Revealed, and it implied that, during ancient China, cavalry was developed in response to the use of chariots (after having learned of the effectiveness of cavalry from the Mongolians, the film mentions). I always assumed that chariot technology came after the use of cavalry, since cavalry seems like a direct application of a horse.
Were chariots heavily used before cavalry in ancient China? If so, why?
Yes, chariots were used in China before cavalry and they learnt cavalry horsemanship from nomads. Not Mongols because there was no Mongols in that time, but other nomadic tribes. Chariots offers better stability. It's easier to shoot arrows and fight on a chariot than on a horse, especially before invention of saddles and stirrups. Besides, ancient Chinese wore robes, not trousers, it's hard to ride a horse with robes. Horse breeds also played a role, China was often lack of good horse breed comparing to nomadic tribes on the north.
Chariot was the main mode of fighting with horse in ancient China up until Warring States period. Chariots were status symbol of a state, defining the state power. Charioteering was a highly prized skill, it was included in six arts for perfect gentleman.
In Warring States period, a king of Zhao realised the advantage of light cavalry and gave order to adopt nomadic cavalry trousers and horsemanship. But even during Qin and early Han dynasties that followed Warring States, China cavalry was still worse than their neighboring threat, the nomadic Xiongnu. The Han emperors realised their inferiority didn't dare to attack into Xiongnu's steppe. Emperor Wu of Han put more effort in his cavalry training and seek a better breed of horse from Ferghana. Only after China got better cavalry, they could defeat Xiongnu. That illustrated how far the difference on horsemanship skills between Chinese armies and nomadic armies in that time.