It does explain similarity of words for familial relations, but not for things like "nose". Babies don't randomly say "nose", or something that sounds similar, while pointing at their nose.
There are also cognitive development constraints, of course.
The question is, from where do we get the correlation? If it's physiological or neurological, then there's nothing to support fanciful claims of innate language models. It's the cognitive constraints that can support fanciful claims, but you have to first show that it's those constraints, rather than the physiological or neurological constraints, that are controlling word choice.
For everything else, not so much.
And especially considering that the researchers "reduced all sounds to 34 distinct consonants and 7 vowels." (Consider that the more we reduce and simplify the vocalizations the more likely we should expect correlations. For example, if we simplified all sounds to grunts we should expect tremendous overlap.)
The correlation of the corpus of 100 words isn't particularly surprising given what we already know about language development. The null hypothesis was that there'd be no correlation whatsoever (i.e. the sounds would be completely random), which I don't think anybody could reasonably expect.