This is because the introduction of Chinese characters also brought with it the introduction of many Chinese loan words which were then adapted to the sound sys

This is because the introduction of Chinese characters also brought with it the introduction of many Chinese loan words which were then adapted to the sound system of Korean (ditto for Japanese). However, words that already existed in native Korean and Japanese were simply assigned 漢字 (Hanzi/Kanji/Hanji) based on their meanings -- this is one of the reasons why in Japanese, Kanji usually have many more readings than the corresponding Chinese Hanzi (Japanese adopted the Chinese onyomi, but they also had to have their own kunyomi readings which were just reverse-engineered based on the existing Japanese word and the assigned Kanji writing).