Unless Chinese is somehow unique among all human languages, this isn't true. Chinese would be just as intelligible if written in a phonetic script (like Pinyin) as it is when written using the characters.
Now, it would be an incredibly shocking transition for Chinese people who have already spent their entire lives writing with characters. However, after the transition to Pinyin, especially for young people who wouldn't ever learn the characters, written Chinese would still be perfectly understandable.
That being said, I don't favor replacing the characters, because the transition would be extremely difficult and because the characters are very culturally important to China. They've been in use for a good 3000 years, and people are very attached to them. Phonetic scripts are technically superior, but the cultural and practical arguments for sticking with the characters are still stronger.