They sincerely believed that when (e.g.) Cantonese speakers read and wrote, they were not wholesale translating from and to Mandarin, but simply, when writing, transcribing Cantonese using the universal ideographic calligraphy.
In fact, writing in dialect languages is not taught. The extremely elaborate system dictating which of usually several, often many syllabary characters that sound identical must be used in writing a word in Mandarin (very commonly mistaken for ideographic writing) cannot work for the other sinitic languages.
(Sinitic languages admit about 1200 distinct syllables, but the syllabary writing system uses many times that number, so many necessarily sound alike. Mandarin speakers are taught that the characters are not merely syllables with attached historical rules, but ideograms that represent distinct thoughts. (Numerous just-so examples are used to support the notion.) This has often led to belief that ancient documents using the characters could be read and understood without deep knowledge of the actual language and world of the writer, resulting in, at best, comical translations.)