That very long and thorough page explains the science, explains why it isn't exact, and does so in a surprisingly approachable manner. However, people seem to judge long, technical explanations as prima facie evidence for the "scientific" soundness of the product, and consider that something which is "scientific" is by definition accurate and reliable. Unhelpfully, that article buries the discussion about precision and recall at the very end. But even if they lead with it, it's perhaps too much to expect them to educate customers on statistical meaning, such as that even 95% precision means 1 out of 20 classifications will be wrong, and thus that it's entirely unsurprising for the journalist to have been misclassified as Middle Eastern.[1]
[1] Though, there's also little evidence that they were misclassified, given the population flows between parts of Italy and other areas around the Mediterranean. Just because their family members came out as all Italian does not imply that the journalist's family lacks Middle Eastern ancestry.