Korean has just as many homophones and used to be written with characters like Japanese. Now they are doing just fine with their phonetic writing system.
Also Korean avoids many homophones thanks to it's 10 vowels. Japanese has 5.
A bit like English "packed" being written with "-ed" even if it sounds identical to "pact". Helps disambiguation.
(Actually, come to think of it, it's rather analogous to the Japanese way of maintaining the same Kanji while the suffix changes.)
Koreans did have an old writing system[1] made of Chinese characters, where some were used for meaning and others were used to denote Korean suffixes with a similar sound (kinda like how Hiragana started out, I guess). But it eventually died out.