Well, first of all I think that, contrary to what you think, most people do know that Chinese refers to mutually incomprehensible dialects.
The definition of "dialects" is that they are intercomprehensible. If they're not, then they are separate languages. Obviously there are borderline cases such as Czech and Slovak, Danish and Norwegian, or Finnish and Estonian, which require increasingly substantial effort for intercomprehension as we move forward on that list, yet are still qualitatively easier than a truly foreign language. And there are language continuums, in which neighboring pairs are easily understood but the ones at the end are considerably harder, such as the geographical spectrum from Berlin German to Amsterdam Dutch.
But the languages of China do not fall into either category. It is a major feat for a Mandarin speaker to learn
Fujian hua or vice versa. These are not dialects, they are distinct languages. The fact that they use the same words in the same order, most of the time, makes this a very interesting subfamily of Sino-Tibetan for linguists, and possibly unique among the world's language groups because their evolution was so profoundly shaped by the technology of writing. But it doesn't change the fact that they are not intercomprehensible by any stretch of the definition, and therefore are not dialects. You could lock a person from Beijing up with a person from Hong Kong for three months and they would be no closer to understanding each other unless they found a way to write on the walls. A Czech and a Slovak will be speaking pidgin in a day and chatting comfortably in a week.
As for "most people," you of course are referring to linguists, anthropologists, sociologists and political scientists. The other 99.999% of Americans believe that all Chinese people speak dialects of a single language just like we do. Just ask any dozen of them. Virtually everyone I ever knew who walked through Chinatown with me asked, "So what are these people saying?" When I said I had no idea because they were speaking Cantonese, they replied, "Oh come on now, Fraggle. You've been studying Mandarin for years and you even begged your Chinese girlfriend and all her friends and family to speak it with you. Surely by now Cantonese is no harder for you to understand than Scots English."
However, this is by no means a justification for having a logographic writing system in the 21st century. Many other languages come in mutually incomprehensible dialects too (e.g., Indian languages . . . .
You missed the point completely. Mandarin, Cantonese,
Shanghai hua and every other language of China
use the same words in the same sequence (mostly). By using logograms everyone writes so that everyone else can read their writing. (They write a little formally so the differences practically disappear, but every written language is slightly more formal than its spoken version.)
This cannot be done with Hindi, Punjabi and Bengali (which are also separate languages, not dialects of one language). They have diverged not just in phonetics but in vocabulary and syntax, just as English and Dutch have, or Italian and Romanian, or Ukrainian and Croatian. They don't use the same words in the same order, so logograms would not help them understand each other.
. . . . Arabic, etc.) and none of these languages solved the problem by doing away with (real) writing-- sorry but this argument makes very little sense.
As for Arabic, those are dialects of a standard language. I don't know how wide a spectrum the dialect continuum covers, so I can't say whether a Moroccan can understand an Iraqi. But a Libyan can understand an Egyptian, an Egyptian can understand a Palestinian, and a Palestinian can understand a Jordanian. Because of the influence of Islam throughout the Arabic-speaking nations, a standard dialect of Arabic has been preserved that does not represent the colloquial speech of any community, and virtually everyone can understand it (especially in the age of radio and TV). This is what is spoken when people from different countries come together, and virtually all written Arabic is a transcription of this preserved standard dialect.
You haven't got precise definitions of "language" and "dialect," so without a clear distinction between the two you're being confused. It's a big and important difference.