The problem isn't Chinese in general, but the written register of Chinese specifically, and all the texts already using it. If you've ever looked at a verbatim transcript of a spontaneous conversation, you'll probably have noticed how much of an incoherent mess spoken language can be. Written speech is much tidier, because people have the opportunity to be very deliberate in what they write down.

For Chinese in particular, that means that people can be much more succinct in writing than orally, because the characters provide additional disambiguation beyond just indicating the pronunciation. Any purely phonetic writing system would require everyone to readjust their standard of what kind of sentences are comprehensible.