One kanji may have more than one way to read it. For example the character 木, meaning “tree” may be read as moku or ki. Japan adopted its writing system from China from the fifth century. The Japanese language is unrelated to Chinese, and the two languages had no common vocabulary originally, but Japanese had no written form. Chinese pronunciations such as moku were introduced to Japan together with the character. However, Japanese already had a word for “tree”, ki. Both the Chinese and the Japanese pronunciations ended up being used to mean “tree”.
The readings like moku, which come from Chinese, are called on-yomi (音読み), and readings like ki, which are original to Japanese, are called kun-yomi (訓読み). The on-yomi readings tend to be used in words made from more than one kanji, called compounds or jukugo (熟語), for example mokuzai (木材), meaning “wood”.
Most Chinese characters came to Japan between the fifth and ninth centuries. The on-yomi is an approximation of the Chinese pronunciation of the character at the time it was introduced. Some kanji were introduced more than once from different parts of China at different times, and so have multiple on-yomi readings. For example the 行 kanji has on-yomi readings gō, gyō, and an.
The most common form of on-yomi readings is Kan’on (漢音), which come from the pronunciation during the Chinese Tang Dynasty in the 7th to 9th centuries. Go’on (呉音) readings, common in Buddhist terminology such as gokuraku (極楽) “paradise”, came to Japan during the fifth and sixth centuries, from the pronunciation of the historical Wu region of China, in the vicinity of modern Shanghai. The less-common Tō’on (唐音) readings, which occur in some words such as isu “chair” or futon come from the pronunciations of later Chinese dynasties, such as the Song and Ming Dynasties, covering readings adopted from the Heian era (794-1185) to the Edo period (1603-1867). In addition, there are Kan’yō-on (慣用音) readings which are mistakes which became established.
In addition to the variety of Chinese-derived readings, a kanji may have several native Japanese kun-yomi readings when a kanji used for more than one meaning. For example, 行 has two kun-yomi readings, iku and okonau. Different kun-yomi can usually be distinguished by kana placed after the kanji, called okurigana. Iku is written 行く, with the kanji followed by く, and okonau is 行う, with the same kanji followed by う.
https://www.sljfaq.org/afaq/kanji-pronunciation.html
And that is the reason why we struggle with it .
why did not the Japanese people have their own writing system ?
Exactly why they didn’t have their own writing system I think is unclear, I remember hearing something about basic record keeping being done in other ways than writing, and that once there were important things to write down the japanese pretty much drew on the to them closest available writing system. My old japanese teacher back at my university once lamented the fact that Japan was located so close to China instead of a culture that used a phonetic writing system so that they had to cut and paste together a writing system of their own. Now, after the fact, the japanese writing system and the language itself is so shaped by the chinese characters that it would be very impractical to take them out.
But you may know that korean folks used the Chinese characters for some time before they bring the one they have now ,which is consider one of the easiest phonetic writing system . So it is not like they can not it is more like a legacy .
I know very little about korean, but as you say they used chinese characters up until recently and then switched to a phonetic alphabet. I don’t know how common homophones are in Korean, and how this is handled in the modern writing system. For japanese, there are so many words, or parts of words, that are pronounced in the same way, that having a writing system where each character represents both a meaning and a sound becomes more necessary. It might not seem necessary at first, but when reading more academic or technical texts it becomes more and more important.
I don’t know about Korean either but Japanese would be a nightmare to read without Kanji. This becomes apparent after getting past the beginner level. I think this has to do mostly with the large number homophones and that Japanese tends use a bigger selection of words than English on a daily basis.
It’s certainly possible to write Japanese phonetically with hiragana and katakana, as you probably know. Japan has used all three alphabets for most of its written history.