I do agree that the informational basis of cultural transmission is an important concept - and that we need terminology to describe it. However, we have some fine terminology developed for that specific purpose: the terminology of memetics: "memes", "memeplexes", "meme pools", etc. Memes are the informational basis of culture. However there's more to culture than memes. There's meme products, namely: artefacts and behaviours. If you "spend" the word culture on its informational aspects, you need new terminology to describe meme products and the union of memes and meme products. I think it's best to call the umbrella category containing memes and meme products by the term "culture". Including artefacts and behaviours seems more consistent with everyday usage - and with historical usage of the term, dating back to Tylor in the seventeenth century.
This use of terminology to distinguish cleanly between the informational and non-information aspects of culture is one of the nice things that memetics inherits from evolutionary biology. There there's "genes" and "gene products". "Genes" refers to the heritable information, and "gene products" are everything else affected by them. The terminology of memetics applies this exact same split to the cultural realm.
From the perspective of culture-as-information, the term "gene-culture co-evolution" makes sense. However, from my perspective, it's an oxymoron. You can have "gene-meme co-evolution" or "organo-cultural co-evolution" - but "gene-culture co-evolution" mixes up different types of thing - and sounds stupid.