In fact, they calculated that words uttered at least 16 times per day by an average speaker had the greatest chance of being cognates in at least three language

In fact, they calculated that words uttered at least 16 times per day by an average speaker had the greatest chance of being cognates in at least three language families. If chance had been the explanation, some rarely used words would have ended up on the list. But they didn’t. I don't think that this logic is valid. It assumes that if the languages had had no common ancestry that any random word would be similarly likely to match between the languages. I'm no linguist but in my experience the most fundamental and commonly used words tend to be shorter, often composed of a single syllable. It also makes sense that the more fundamental words would be created early on and be given simple sounds while later words would be more complicated by necessity. If this is the case then you would expect more random coincidences within the cores of different languages than outside of them. This is also consistent with the fact that most of the examples in the article are very short: thou, give, hand, bark, spit, worm. I think that this would need to be taken into account in the statistical analysis before attempting to draw any conclusion.