This part seems to have some presuppositions that need to be corrected:
Is there a non-negligible chance that this correct guess could be
actually caused by the shared root of the proto-human language
First, it is possible that there was a single earlier language from which all current languages derive, but it is also possible that there is no single common ancestor to all human languages. Let's just assume that there was one proto-language, then what would it mean for a shared root to "cause" significant recognizability in words? We can take cases where having a "shared root" does affect guesses. Speakers of Germanic languages can guess the meaning of the word "hand" in another Germanic language because the words of Dutch, English, German, Danish, Swedish, Norwegian and Icelandic are derived from the same root of proto-Germanic. Likewise, speakers of Bantu languages can guess the meaning of the verb "to cultivate" in another Bantu language, because they nearly all have the same word pronounced [kʊrɪma] or something close to that. In such cases, you can say that ability to guess meaning correctly is caused by the shared root of Proto-Germanic or Proto-Bantu. But that implies something very weird: that speakers of English or Norwegian have some kind of mystical access to a fact of a language spoken millenia ago, which nobody alive now has ever heard.
What allows speakers of Germanic or Bantu to make these "educated guesses" is not the fact that the modern words historically come from a common root, it is the fact that the modern words look very similar (which itself is because they derive from a common foot, and because the sound changes that affected these particular words happened to have not changed the shape of these roots all that much).
When the modern words are not similar in shape, people are unlikely to correctly guess the meaning of a word in another language, thus speakers of Indo-European (who aren't bilingual and haven't studied Indo-European linguistics) will not guess that koun, ci, hound, chien, šuõ, šun, sobaka all mean the same thing. The root *gʷʰen is common enough in Indo-European languages, but people generally can't guess that because the contemporary manifestations are so diverse (defense, gìnti, gon, gjanj, φόνος, ganem). Even if a pair of words does historically derive from a common source, that fact doesn't cause people to correctly guess the meaning of the words.
There is a very small set of words whose meanings are cross-linguistically guessable at a somewhat significant rate, especially "mother", "father", "dog", "chicken". The problem is that these words also have a tendency to be irregular from the perspective of historical sound changes – they tend to resist regular changes, as though there is some sound-symbolic force that keeps them exceptionally close to the standard of mama, dada/baba, bu, kuk, to the point that it would be impossible for such similarity to be preserved even if Proto-Human had these roots which were passed down to most modern languages. It's that sound-symbolic force, and not a common root in a language possibly spoken 100,000 years ago, that causes good-guessing behavior for these words.