Why didn't Zheng He sail to discover America?
There is a whole theory that they did.
Zheng He was made a number of voyages commissioned by the Emperor Yongle after 1405 well ahead of the Columbus voyages.
The Chinese used very large ships capable of long ocean voyages. Many of the ships did not return home so it was theorized that they may have ended up in America or other places. This is not generally accepted.
There have been some theories of ancient contact. DNA evidence of Chilean chickens having genetic similarities to Eurasian chickens has been presented as evidence of this. The following site discusses some of them:
There was a change in the Emperor and a change in the administration under the Emperor. The Scholar-bureaucrats were more concerned with internal matters particularly when the ships failed to find any civilization equal to that of China at the time. China viewed everyone else as vassal states and began to lose interest.
China became a self contained and independent Empire without the need of outside trading partners from minor nations.
China continued with these views until the Opium Wars.