One classic finding in linguistics is that languages with lots of morphology tend to have freer word order. Latin has lots of morphology and you can move the verb or subject anywhere in the sentence and it's still grammatical. In a language like English syntax and word order and word choice take on the same role as morphology.
Inflected languages may indeed have more information encoded in each token. But the relative position of the tokens to each other also encodes information. And inflected languages appear to do this less.
Languages with richer morphology may also have smaller vocabularies. To be fair, this is a contested conjecture too. (It depends a lot on how you define a morpheme.) But the theory is that languages like Ojibwe or Sansrkit with rich derivational morphologies and grammatical inflections simply don't need a dozen words for different types of snow, or to describe thinking. A single morpheme with an almost infinite number of inflected forms can carry all the shades of meaning, where different morphemes might be used to make the same distinctions, in a less inflected language.