That's a vast understatement. Adopting a writing system that would render everything written in the formal register incomprehensible isn't exactly a tenable position.
But granted, the transition took a while and was not just a matter of habit.
Having said that, why could one not publish important classic texts in a phonetic vernacular version, and leave the study of the originals to those with the inclination (just as most people in the West do not learn Hebrew, ancient Greek, and Latin; yet can find versions of the Bible, Euclid, Seneca, and Newton that they can read if they so desire.)
It's unclear to me how far that extends. (This issue seems intertwined with the issue that many speakers of other Chinese "dialects"/"languages" also use Standard Written Chinese (ie Mandarin), btw.)
It is possible to write some register of Chinese in phonetic writing, though - that's shown by Dungan (written in Cyrillic), by the experiments in the 30s (quoting Mao again: "we have begun experimenting with Hsin Wen Tzu—Latinized Chinese. It is now used in our Party school, in the Red Academy, in the Red Army, and in a special section of the Red China Daily News."), etc.
First, Mandarin has no inflection (love/loves), which allows for writing in characters in the first place. Then, Mandarin has a fairly impoverished phonetic system, allowing for only about 400 different syllables ignoring tones, or 1200 or so when distinguishing tones (while English for example has some 7000 or so, without tones). This leads to many homophones, which can be distinguished by being written with different characters. Or so the traditional justification goes (I don't think much of that argument, because clearly people can speak to each other just fine, without having to clarify which word they mean - disambiguation happens by longer words (two or more syllables), which you could just write down. What happens is that in writing, you only need to write parts of a word: spoken out loud it would be ambiguous, but the chosen character disambiguates it, and thus people conclude that writing wouldn't work without the characters.)
However, arguably it's just a matter of habit (and resistance on the part of those that have already put in the time and effort to learn the characters). There are, in fact, Sinitic languages written alphabetically, eg Dungan [2].
The whole issue is very politicised, as you can see with the animated discussion here and the vigorous resistance to the initial comment (which was a bit unfortunately phrased, in my view, but fundamentally sound: Chinese is a very hard language to learn, mostly by virtue of its archaic writing system, and that is arguably somewhat deliberate.)
[1] https://en.wikipedia.org/wiki/The_Chinese_Language:_Fact_and...
[2] see the articles linked from here: http://languagelog.ldc.upenn.edu/nll/?p=40417