This isn't really a full answer to your question, but I think there's a non-trivial amount of truth to it.
When you look at the subject of operads, what kind of diversity do you see?
I prefer the topological category so I'll list those in the topological category that come to my mind:
Discs operads (equivalent to cubes)
Framed discs operads (a semi-direct product of $SO_n$ with discs.
Free product, tensor product operads...
Cacti operads (different ones are equivalent to discs and framed discs....)
McClure and Smith's operads, which are equivalent to cubes / cactus operads.
Overlapping cubes operads (my terminology), which are equivalent to cubes operads.
What people in the embeddings of manifolds community call the Kontsevich Operad, which is equivalent to cubes / discs operads.
The operad I call "the splicing operad", which operad" in dimension $3$, is a free product of free operads with a semi-direct product of $O_2$ and the $2$-cubes operad...
Anyhow, if you look at the above, what do you see? Free operads (decorated trees) and cubes operads and various natural constructions. Moreover, some of these construction produce "more of the same" -- the tensor product of cubes is equivalent to cubes, for example.
An impression you could take out of this story is that operads perhaps have very few "core types" but a prolific number of variations on those "core types". One possible exception to this is the high-dimensional splicing operad may not be reducible to these "core types" but that is not clear at the moment.
So perhaps what you see isn't a physics thing, it's an operads thing?