The linguist states that the two groups don’t disagree over the idea that some paintings aren’t art. The second group, like the first, believes some paintings should be dismissed as non-art.

There’s nothing showing that an idea is a “commonly accepted” reason for accepting another claim. Nor is there any independent evidence for the hypothesis that the two groups don’t disagree over the meaning of “art.”

The linguist isn’t arguing that the referenced text is actually true. He’s simply pointing out that the two groups agree that not all paintings are works of art. We don't know that the linguist herself believes that not all paintings are works of art.

The linguist says nothing about aesthetic merits. She also doesn’t generalize about the types of disputes that ensue from claims about the nature of art.

The referenced text shows that both parties accept that not all paintings are art. The linguist uses that point of acceptance to draw the conclusion that the two disputing parties aren’t disagreeing over the meaning of the word “art.”