How would you replace the Kanji writing system?

Binary. It'd give people a writing system that's actually worth complaining about.
Or, even better, the wave pattern of a word in Japanese; that way, you'd get the pitch accent in there too, so no one can complain about there being too many homonyms. No mathematical models allowed (it wouldn't really be a new writing system then), everyone will just have to get really good at squiggling on very long pieces of paper and get used to handling scrolls again.