I am pretty sure that there is no "decent" study of the question, i.e. one that is non-anecdotal and is well-balanced in the languages considered, so it's unsubstantiated; but the feeling probably could be born out, within the margins of what's clearly establishable in morphology. For example, the majority of roots in Bantu languages are C(V)V(N)C, though there are some roots like *kamat "hold" which seem a bit longer: but this is usually analyzed as a CVC "core" root plus a non-productive extension -at. Austronesian languages have a considerable share of longer roots that look reduplicated, so one might reduce with "roots" to a true root plus a reduplication morpheme. If you have weak criteria for establishing what is a root, it's easier to eliminate longer roots.
The other problem has to do with claims about "on average", as a claim about human language . On average, languages are Niger-Congo or Austronesian, so if it turns out that most Niger-Congo and Austronesian roots are 5 phonemes or less, you'll probably get the desired result. It is probably true that in most Niger-Congo languages you have one root syllable with maximally CVC struction, and Austronesian roots tend, as far as I know, to be shortish.