Dialects
I think: the idea that Minnan (Taiwanese, Hokkien), Cantonese, Shanghai and so on are dialects of Chinese (putonghua, kuoyu, mandarin) is a political construct aimed at underpinning the idea of the Chinese state and people as one, a single unity, and a unity that includes Hong Kong and Taiwan. In the case of Taiwan, it was one of the methods the KMT dictatorship relied on to suppress the Minnan language and a distinct Taiwanese identity. It is meant as a way to keep the Chinese state together. If we really were talking about dialects, then I think Minnan/Taiwanese is the older of today's Sinitic languages, and in that case Putonghua/Mandarin would be a dialect of Minnan. But what are the odds that we will hear Hu Jintao say that he speaks a dialect of Taiwanese?
And as for Minnan/Taiwanese being a fang1yan2/dialect: by the late Qing dynasty, the word fang1yan2 still meant "foreign language", and language, interpretation and translation schools were called "fang1yan2 xue2tang2." Further, there are many Taiwanese words that can't be written using Chinese characters. One good example is the word "kiu," normally written "Q" here in Taiwan and meaning "chewy." My Taiwanese teacher doesn't know how to write it, and when I checked three different Taiwanese/Minnan dictionaries, they all gave a different character, two of them made up since there is no existing character, and one using a far-fetched existing character (the Minnan dictionary from Xiamen, of course, which goes so far in the fiction that Minnan is a dialect of Mandarin that the whole dictionary is arranged by Putonghua pronunciation, not by Minnan.).
There actually seems to be quite a few Taiwanese words that cannot be written in Chinese script. One study I read, which also referred to a couple more (this is one of them), put the number at about 20-25%. As mentioned above, I have certainly encountered quite a few such words in my stumbling Taiwanese studies, although the percentage at my limited level seems to fall below that number (maybe character-less words are more common in more advanced language, or maybe the estimates are off). Part of the remaining are written using homophones where only the sound, not the meaning of the character, is relevant. The same study suggested that about 25% of Taiwanese characters is written in this way, which, if correct, would leave only about 50% of Taiwanese words having characters where the sense of the word and the character coincide. But that is true of many languages -- there are many words in European languages that coincide with Latin and Greek words without prompting anyone to claim that English is a dialect of Latin.
In short, I don't think Minnan or Cantonese are dialects of Chinese any more than English or Swedish are dialects of German. They, together with mandarin, are today's sinitic languages that all developed from a pro-sinitic language spoken in archaic times, in the same way as German, Swedish, Dutch, English and all the other Germanic languages developed out of a proto-Germanic language long since dead.