Most people will say it is because there are too many homonyms in the Chinese language. But when I realize that homonyms do not bother people very much in oral communications, I begin to question this answer. As a matter of fact, Vietnamese writing system has been successfully romanized. They have a lot of homonyms too, but they are doing fine.
Ever since Sejong the Great invented the Korean alphabet in 1446, Korean vocabulary has become a superset of the Chinese vocabulary. In other words, hangul has no problem with Chinese homonyms.