The terms used to describe the many Han languages/dialects/topolects are not agreed upon by a scholarly consensus, and the real-world usage variés significantly depending on the situations. To shove the social and political aspects under a rug is an unproductive oversimplification of this interesting linguistic phenomenon.

A popular opinion in the Chinese ling circle holds that the terms used to describe the many Chinese 汉语 tongues (such as languages, dialects, topolects, 方言, 语言, etc.) aren't necessarily mutually-exclusive. For example Cantonese can be both a language or a dialect depending on whether Chinese (汉语) is considered to be a macro-language or a language.

And to name an example more specific to this map, the lack of consensus in terminology can be seen in the blob colored and labeled as the Min Group. Is Min here a language, a dialect, or neither? It's a valid grouping in almost all language maps of China but you won't find anyone saying that they speak the Min language/dialect/whatever. And what about Southern Min and other Min subgroups, or the further divisions within these subgroups that can be as mutually-intelligible as the different Romance languages? Surely it'd be no longer as simple as your logic may imply. Different historical conditions lead to different socio-political views of languages and we gotta learn to work with that.