Lev,
The "desire to see something 'mental'...emerge as the principal element in the structure of the Universe" is as old as the theistic religions. See my paper to see how Bishop Berkeley hoped to accomplish just that as he placed physics on the path of merely describing the contents of the observer's consciousness, as if no physical Cosmos existed. That path led to Relativity and QM. The Cosmos was replaced by the observer's information.
You ask, "Which new non-spatial form(s) of 'data' presentation will reveal the recently invisible and allow us to understand adequately the formative processes in Nature?". We already have it, it is called "natural philosophy". It is the use of our full intellectual capabilities to reach beyond the "information" of your conscious experience and create theories about what the Cosmos is made of, what causes all its processes, and how it evolves its complex structures including ourselves.
Your ETS formalism may prove to be a useful tool of philosophy. It will still, like mathematics, require human philosophical intelligence to abstract from reality and produce the symbolic representation, and then again to apply the representation to any particular real situation.
Henry