While this might sound sensible, few definitions of the term "meme" make any mention of the point. In most cases, the lack of a specification of linked parts allows memes to consist of components that lack memetic linkage.
The conception of a "cultural variant" in academia also typically lacks any specification of linkage. I've long claimed that the term "cultural variant" - used by some in academia - is a synonym of "meme", but I'm now starting to doubt that: "meme" now seems to have stronger implications of linkage than the term "cultural variant" does.
The main downside of having memes being defined as having linked components is that it seems to make their definition more complex. It would no longer be possible to say that memes are what meme frequency analysis studies - or that memes are small, inherited pieces of culture. Simple definitions are good. It also seems undesirable if linkage-related issues complicate the issue of what counts as a meme. This whole issue seems non-trivial to me.