I don't understand your objection to the slicing procedure.
The central assumption is that for a reasonably physical spacetime there exists a unique maximal globally hyperbolic solution to the field equations, justifying the use of the initial value formulation of General Relativity. [aside: [1]]
Once we do that we can take any arbitrary spacelike hypersurface Σ and in ADM-ish vocabulary observe the compact closure conditions (Σ has a spacelike geodesic from every p in Σ to every other p in Σ; every maximal spacelike geodesic has a nonempty open segment in Σ; every maximal timelike geodesic intersects Σ), the "thin" condition (no nomepty segment of a null geodesic is found in Σ), and a variety of causal conditions (e.g. no timelike geodesic connects points in Σ; no maximal timelike or null geodesic intersects Σ more than once).
One can translate these features to your cones-only view. On a Minkowski diagram with p at the origin, Σ can have any shape or orientation compared to the light cones as long as p -- and nothing else in either of p's light cones -- appears on Σ. We can do the same for any other p in Σ, or we can decide whether the origin of any pair of light-cones is in Σ or not. We can also probably build Σ by hand from a principled choice of available light-cones. (One would be tempted to call that "initial data").
Is Σ physical? Of course not! But it can be used as the basis for physics that describes point-coincidences on other spacelike hypersurfaces, which we can then verify.
> The future light cone is what can possibly be changed based on things we do here and now
In which of your three regions do we find the freedom to do this?
Is here-and-now not fully determined by the "fixed and certain" content of its past lightcone?
Is the "we" part of "what we do here and now" describing a point on the manifold or is it describing a set of points on the manifold? Choose a point somewhere in the middle of "we" -- in which of the three regions are its other points?
I struggle to answer that without thinking about an extended "we" with many entries in Σ (including photons (and other carriers of body forces) bouncing around within or even passing through the extended body). Maybe it's much easier for you.
I wonder how you'd do if you were to think of grossly non-Eulerian (and even non-timelike) observers in a Big Crunch universe. In a Big Bang universe all points are in the future light cone at BB; in a BC universe all points are in the past light cone at BC. At every other point, only some points are in the two lightcones at that point. We get a BC from a time-inversion of our BB->approx de Sitter universe (e.g. for an observer with differently-aligned arrow-of-time). Assuming a long period of matter sparseness, in a BB->BC universe, we also get a ~BC universe if we cut away the expanding part; if we cut away the recollapsing part, we get something that's hard to distinguish from a BB->~dS universe like our own. In a BB->~dS universe I think we'd agree that the future light cones of two observers coincident at p can eventually fully separate. Can we agree that in a BC universe the past light cones of two observers at p can be at least somewhat, or even (practically) fully, non-overlapping (our observers need not be either "Eulerian" comovers with the metric contraction, and can even move lightlike, and there can be a cosmic deflation just prior to the big crunch)?
In an ADM(-like) picture we can fairly straightforwardly use ancestor slices of Σ in the foliation and decide that a pair of photons will be at some point p on Σ. I also think it makes a slice through a spacetime-filling electromagnetic field easier to think about, ditching the idea of bare photons as independent objects. Again it doesn't matter about the timelike direction of the slicing, and one might want to recalculate slicing along some other choice of timelike axis to try to spot coordinate artifacts. However maybe you might object that if we don't have full data for the ancestor of Σ, we're somehow not better off than in the final sentence of your second paragraph.
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[1] Weaker causality, such as stable causality in the sense that arbitrarily small metric perturbations don't give rise to CTCs, can still be used but at the cost of solution uniqueness, and weaker still at the cost of smearing out the Cauchy surface so that initial data is not all at t=0 but in some function defined over 0 >= t >= -V^{-1} where V is the scale of strong causality violation.