There are a few reasons, but mainly a lack of support.

After Korea became a Japanese colony and the Joseon dynasty was completely over, the Korean Provisional Government in exile was founded in Shanghai. It was run as a democratic/representative government with a president and prime minister. Members of this government were active in overthrowing the Japanese regime. Kim Gu, Syngman Rhee, and Kim Kyu-sik were all active members of the government.

At the same time, what was the royal family doing?

Yi Eun, son of the Gojong, was a general in the Japanese army and he married a cousin of the Japanese emperor. Yi Kang, another son of Gojong, attempted to join the Provisional government, but they weren't enthusiastic about it. He returned to Korea and supported the Japanese. Kang's son, Yi Wu, joined the Japanese military and was killed as a Lt. Colonel in the bombing of Hiroshima.

There are plenty of other examples, but Korea collapsed and was taken over by the Japanese because the monarchy had grown weak. After Japan took over, they continued on as puppets. Compare that to the Provisional government that was actively fighting the Japanese and it's easy to see why there wasn't much support for bringing them back.