Yes of course, that question was just for me. Thanks.

I added references and a remark on cofibrancy issues. If we require all the finite intersections to be cofibrant, that's enough to recover your argument. Also, what is your intuition for quickly seeing what the Moore normalization is? I never had intuition for this as the definition is somewhat involved, and tend to stick with the alternating face map complex for simplicity. But this is something I'd like to have a feel for: just looking at it and saying: "ahh the normalization is this which is much nicer to look at."

Possibly. Each term in the first page should be able to be described as the homology of the simplicial replacement of this simplicial space composed with the cohomology functors, which in some sense contains these tuplewise intersections; but it doesn't look as nice as this description. It contains too much information, and it is not obvious (except for maybe the two cover) that on passing to the second page that extra information gets wiped out and your left with the nicer description above.

Here's an alternate approach: given a spectrum, we can construct the homotopy colimit spectral sequence associated to a diagram, which in this case we choose to be this simplicial space you defined above. Since the homotopy colimit of that simplicial space is weakly equivalent to $X$, we get the convergence to the homology of the space $X$. The second page can then be described in terms of the derived colimit functors, and the spectrum. Definitely not as concise a description as yours, at the second page. Actually, this is exactly @DenisNardin s answer. I may edit this wiki later.

The theorem you reference presupposes the simplicial space is Reedy cofibrant, which I don't think this simplicial space you define is, generally. You might be able to recover something with a cofibrant replacement functor. Also, Dugger shows that the geometric realization of this simplicial space is always weakly equivalent to $X$, which should give the desired isomorphism.