You might enjoy

https://www.ams.org/journals/bull/2004-41-03/S0273-0979-04-01026-2/S0273-0979-04-01026-2.pdf

which explains the role that the resulting problem (representing homology class of manifolds by submanifolds/cobordisms) played in inspiring the work of René Thom on cobordism, stable homotopy theory, singularity theory...

Here are two more closely related results in the same circle of ideas. The first one gives a description (a kind of fusion of Dold-Thom and Eilenberg-Steenrod) of homology purely internal to homotopy theory, and the second explains how homological algebra falls out of infinity-category theory:

• Consider functors E:S_* --> S_* from the infinity-category of spaces to itself which commutes with filtered colimits, carries pushout squares to pullback squares, sends the one-point space to itself, and the 0-sphere (aka two points!) to a discrete space. Then A=E(S^0) has a natural structure of abelian group, and E is an (infinity-categorical version of) the Dold-Thom functor and satisfies pi_n E(X)=H_n(X,A) (reduced homology). In particular, E(S^n) is an Eilenberg-Maclane space K(A,n).
• The category of functors E:S_* --> S_* satisfying all the properties above except the one about E(S^0) being discrete is a model for the infinity-category Sp of spectra, i.e. the "stabilization" (in a precise categorical sense) of the infinity-category of spaces. From this perspective, the functors from the previous points are called the Eilenberg-Maclane spectra HA. Moreover, the infinity-category of spectra has a symmetric monoidal structure (the "smash product"), HR is naturally an algebra object for this structure whenever R is a ring, and it makes sense to talk about the infinity-category LMod(HR) of left HR-modules in Sp. Then LMod(HR) is equivalent (essentially by a stable version of the Dold-Kan correspondence) to the infinity derived category of left R-modules D(R). In other words, for homotopy theorists, (chain complexes,quasi-isomorphisms) are just a funny point-set model for HR-module spectra! 

All of this is discussed in the first chapter of Lurie's Higher Algebra, except the last point which is not completely spelled out because monoidal structures and modules are only introduced later on.

I should point out that this perspective is largely a reformulation of the results you already mentioned, and in themselves certainly do not bring new computational techniques for singular homology. However they show that 1) homological algebra comes out "structurally" from homotopy theory, which itself comes out "structurally" from infinity-category theory and 2) homological algebra (including in more sophisticated contexts than just abelian groups, e.g. dg-categories), homotopy theory, sheaf theory... can be combined inside of a common flexible categorical framework, which elegantly subsumes previous point-set level techniques like model categories. 

All the frames you are mentioning are good for intuition. I would say the deepest one is 4. and that everything falls into place cleanly once you formulate things in the language of infinity-category theory (at the price of a lot of technicalities to establish the "right" language). For example, 

• singular homology with coefficients in A can be characterised as the unique colimit-preserving infinity-functor from the infinity-category of spaces/homotopy types/infinity-grpoids/anima to the derived infinity-category of abelian groups which sends a one-point space (equivalently any contractible space) to A[0].
• The derived infinity-category of abelian groups is itself in some sense the "(stable presentable) Z-linearization" of the infinity-category of spaces, although this is more tricky to state precisely and I won't try to do this here.

Which formal properties of the KL-divergence do the proofs of your result use? It could be useful to make them all explicit to help generalize to other divergences or metrics between probability distributions.

Well, I can certainly emphasize with the feeing that compromising on a core part of your identity is threatening ;-)

More seriously, what you are describing as empathy seems to be asking the question:

 "What if my mind was transported into their bodies?"

rather than 

"What if I was (like) them, including all the relevant psychological and emotional factors?"

The latter question should lead feelings of disgust iff the target experiences feelings of disgust.

Of course, empathy is all the more difficult when the person you are trying to emphasize with is very different from you. Being an outlier can clearly make this harder. But unless you have never experienced any flavour of learned helplessness/procrastination/akrasia, you have the necessary ingredients to extrapolate. 

Historically commutative algebra came out of algebraic number theory, and the rings involved - Z,Z_p, number rings, p-adic local rings... - are all (in the modern terminology) Dedekind domains.

 Dedekind domains are not always principal, and this was the reason why mathematicians started studying ideals in the first place. However, the structure of finitely generated modules over Dedekind domains is still essentially determined by ideals (or rather fractional ideals), reflecting to some degree the fact that their geometry is simple (1-dim regular Noetherian domains). 

This could explain why there was a period where ring theory developed around ideals but the need for modules was not yet clarified?

Modules are just much more flexible than ideals. Two major advantages:

• Richer geometry. An ideal is a closed subscheme of Spec(R), while modules are quasicoherent sheaves. An element x of M is a global section of the associated sheaf, and the ideal Ann(x) corresponds to the vanishing locus of that section. This leads to a nice geometric picture of associated primes and primary decomposition which explains how finitely generated modules are built out of modules R/P with P prime ideal (I am not an algebraist at heart, so for me the only way to remember the statement of primary decomposition is to translate from geometry 😅)
• Richer (homological) algebra. Modules form an abelian category in which ideals do not play an especially prominent role (unless one looks at monoidal structure but let's not go there). The corresponding homological algebra (coherent sheaf cohomology, derived categories) is the core engine of modern algebraic geometry. 

BTW the geometric perspective might sound abstract (and setting it up rigorously definitely is!) but it is many ways more concrete than the purely algebraic one. For instance, a quasicoherent sheaf is in first approximation a collection of vector spaces (over varying "residue fields") glued together in a nice way over the topological space Spec(R), and this clarifies a lot how and when questions about modules can be reduced to ordinary linear algebra over fields.

Some of my favourite topics in pure mathematics! Two quick general remarks:

1. I don't hold such a strong qualitative distinction between the theory of group actions, and in particular linear representations, and the theory of modules. They are both ways to study an object by having it act on auxiliary structures/geometry. Because there are in general fewer tools to study group actions than modules, a lot of pure mathematics is dedicated to linearizing the former to the latter in various ways.
2. There is another perspective on modules over commutative rings which is central to algebraic geometry: modules are a specific type of sheaves which generalize vector bundles. More precisely, a module over a commutative ring R is equivalent to a "quasicoherent sheaf" on the affine scheme Spec(R), and finitely generated projective modules correspond in this way to vector bundles over Spec(R). Once you internalise this equivalence, most of the basic theory of modules in commutative algebra becomes geometrically intuitive, and this is the basis for many further developments in algebraic geometry.

There is another interesting connection between computation and bounded treewidth: the control flow graphs of programs written in languages "without goto instructions" have uniformly bounded treewidth (e.g. <7 for goto-free C programs). This is due to Thorup (1998):

https://www.sciencedirect.com/science/article/pii/S0890540197926973

Combined with graphs algorithms for bounded treewidth graphs, this has apparently been used in the analysis of compiler optimization and program verification problems, see the recent reference:

https://dl.acm.org/doi/abs/10.1145/3622807

which also proves a similar bound for pathwidth.