That seems far too simplistic. I'm a native Dutch speaker, and can understand written Afrikaans (spoken can be a bit harder), but it's still clearly a different language with different spelling, grammar, pronunciation, and literature.
I looked up some examples of Appalachian on YouTube, and I can follow it surprisingly easy – much easier than I would have expected after reading this thread – as a proficient but non-native English speaker. It's certainly not a different language; I have more trouble understanding some dialects in my native Dutch.
There is absolutely a grey area where things start to diverge, and this is where politics and the (self-)identification of the people who speak the language(s) can come in to play. Classification is always hard (see also: defining "species" of animals), but a blanket "it's a political question, full stop" is far too strong and hand-wavy.
In this case, it is, nevertheless, correct. There's simply no clear-cut objective boundary here. You can arbitrarily define one as N% of mutual intelligibility, but then your pick of N is itself a political question.
Keep in mind that many of what we today consider distinct languages historically formed a dialect continuum - usually geographic, where you can observe gradual changes as you travel, which may add up to the extent that there's no real mutual intelligibility between the beginning and the end of your path. Are they different languages at that point? If so, what does that make of all the gradual steps in between?
The stark language boundaries that we're more familiar with today are largely due to states aggressively encouraging (or outright forcing) some kind of language standard on what was before a dialect continuum area, smoothing out the differences within it while sharpening them at the borders. This process kicked into high gear with the rise of nation-states, which would often do so for ideological reasons - basically, to force conformity and reify the abstract notion of "one people". But it hasn't happened everywhere to the same extent as it did in Europe, and there are still plenty of places around the globe where clear language boundaries don't really exist.
So I looked it up. Wikipedia (for what that's worth) clearly says that linguistically it is a dialect of American English. The dispute is over whether it is a separate dialect, or part of the Southern dialect.
There are plenty of mutually unintelligible dialects and plenty of mutually intelligible languages.