This is a problem I have thought alot about. I have not seen any of the modern techniques in your list applied to the problem. Part of the issue is that if you represent $\sigma(n)=2n$ as a Diophantine equations in $k$ variables (corresponding to the prime factors--but allowing the powers to vary) then there are lots of solutions (just not where all the variables are simultaneously prime). So the usual methods of trying to show non-existence of solutions just don't cut it. Historically, this multiplicative approach is the one many people have taken, because at least some progress can be made on the problem. My personal feeling is that maybe someday these bounding computations will be tweaked to the point that they lead to the discovery of some principle that will solve the problem. For example, in one of my recent papers, I was led to consider the gcd of $a^m-1$ and $b^n-1$ (where $a$ and $b$ are distinct primes). I would conjecture that this gcd has small prime factors unless $m$ or $n$ is huge. If that happens, many of the computations related to bounding OPNs become much easier.
I have occasionally thought about whether modular forms might say something about this topic (which is why I'm currently sitting in on my colleague's course). Instead of $\sigma(n)$, the `right' function to consider is $\sigma_{-1}(n)=\sigma(n)/n$ and I don't know off the top of my head if it appears in connection with (weakly holomorphic) modular forms. But I know there are some nice techniques about multiplicative functions that decrease over the primes, etc...
I like the simple slogan: homotopical algebra is the nonlinear generalization
of homological algebra. Let me assume that you value and appreciate homological
algebra in the broadest sense as a fundamental, successful and highly applicable tool in many areas of math (otherwise I can't conceive of an argument that would be convincing for this question). At the coarsest level homological algebra is based on the idea of resolutions, i.e. that to perform algebraic operations on objects we should describe them in terms of objects that behave well for the given operations.
Now let's observe that homological algebra is a linear theory,
in the sense that it deals with things like vector spaces, modules over a ring,
and more generally objects of abelian categories. What if your interests
involve more complicated objects that are not linear? for example, rings, algebras,
varieties, manifolds, categories etc? philosophically it still makes sense that
we have much to gain by resolving in some appropriate sense. Homotopical
algebra is the language and toolkit built for this explicit purpose, and with many explicit applications. The $\infty$-language in my mind is just a very convenient and relatively friendly apparatus to understand, navigate and apply this theory.
Some key examples:
$\bullet$ Hodge theory. For me (and I assume many other algebraic geometers) the first
instance of homotopical algebraic thinking I encountered was Deligne's construction
of mixed Hodge structures on the cohomology of complex algebraic varieties, one
of the most powerful tools in modern algebraic geometry. The idea is that the functor "de Rham cohomology" is very wonderfully behaved on smooth complex projective varieties, and
most importantly carries a rich extra structure, a pure Hodge structure. We can take advantage of this for say any singular projective variety if we use the idea of
resolution, in the form of a simplicial object (a convenient nonlinear version of a chain complex) --- we replace the variety by a simplicial smooth projective variety which
is equivalent in the appropriate sense, in particular will produce the same
measurement (cohomology). The existence of such is deep geometry (resolution of singularities) but its explicit applications don't require explicit knowledge of this geometry. It now follows that the singular variety's cohomology carries the appropriate derived version of a pure Hodge structure, namely a mixed Hodge structure.
$\bullet$ The tangent complex. Another seminal circa 1970 application is the
Quillen-Illusie theory of the tangent complex. Again we want to do basic geometry - this time calculus - on a singular variety, or perhaps let's say a commutative ring, so we resolve it in the sense that befits the problem. We like affine spaces for
taking derivatives etc, so if we want to calculate derivatives (tangent spaces) on a singular variety we should resolve it by such --- replace a ring by an appropriate free resolution (this time a COsimplicial variety). This gives us a way to extend the
basic tools of calculus to singular varieties, with many corresponding applications.
$\bullet$ The virtual fundamental class. This is an elaboration on the previous point which is much more recent. We would like now to integrate on a class of singular varieties,
so need a version of the fundamental class. The varieties in question arise as moduli spaces (say in Gromov-Witten or Donaldson-Thomas theories), which means they are relatively
easy to resolve in a natural way (express as a derived moduli problem). As ordinary varieties they are very badly behaved (eg are not even equidimensional) but the derived moduli problem naturally carries a fundamental class.
$\bullet$ In representation theory the key objects
of study are again nonlinear --- associative algebras (or equivalently their categories of
modules). Thus to perform algebraic operations on these algebras we gain much by
allowing ourselves to resolve them. As mentioned above the geometric Langlands program
is one place where homotopical language is extremely useful, but one can find the same issues in studying say modular representations of finite groups (eg the theory of support varieties and stable module categories). More generally Hochschild/cyclic theory, the "calculus" of associative algebras/the fundamental invariants of noncommutative geometry, are natural applications of homotopical algebra. There are many spectacular achievements in this area, one famous one being the Deligne conjecture/Kontsevich formality/deformation quantization circle of ideas. The cobordism hypothesis, in my view one of the pinnacles of homotopical algebra, has among its many facets a vast generalization of Hochschild theory.
Best Answer
One of the closer connections to geometric topology is likely from invariants of manifolds. The motivating reason for the development of topological modular forms was the Witten genus. The original version of the Witten genus associates power series invariants in $\mathbb{C}[[q]]$ to oriented manifolds, and it was argued that what it calculates on M is an $S^1$-equivariant index of a Dirac operator on the free loop space $Map(S^1,M)$. It is also an elliptic genus, which Ochanine describes much better than I could here.
This is supposed to have especially interesting behavior on certain manifolds. An orientation of a manifold is a lift of the structure of its tangent bundle from the orthogonal group $O(n)$ to the special orthogonal group $SO(n)$, which can be regarded as choosing data that exhibits triviality of the first Stiefel-Whitney class $w_1(M)$. A Spin manifold has its structure group further lifted to $Spin(n)$, trivializing $w_2(M)$. For Spin manifolds, the first Pontrjagin class $p_1(M)$ is canonically twice another class, which we sometimes call "$p_1(M)/2$"; a String manifold has a lift to the String group trivializing this class. Just as the $\hat A$-genus is supposed to take integer values on manifolds with a spin structure, it was argued by Witten that the Witten genus of a String manifold should take values in a certain subring: namely, power series in $\Bbb{Z}[[q]]$ which are modular forms. This is a very particular subring $MF_*$ isomorphic to $\Bbb{Z}[c_4,c_6,\Delta]/(c_4^3 - c_6^2 - 1728\Delta)$.
The development of the universal elliptic cohomology theory ${\cal Ell}$, its refinement at the primes $2$ and $3$ to topoogical modular forms $tmf$, and the so-called sigma orientation were initiated by the desire to prove these results. They produced a factorization of the Witten genus $MString_* \to \Bbb{C}[[q]]$ as follows: $$ MString_* \to \pi_* tmf \to MF_* \subset \Bbb{C}[[q]] $$ Moreover, the map $\pi_* tmf \to MF_*$ can be viewed as an edge morphism in a spectral sequence. There are also multiplicative structures in this story: the genus $MString_* \to \pi_* tmf$ preserves something a little stronger than the multiplicative structure, such as certain secondary products of String manifolds and geometric "power" constructions.
What does this refinement give us, purely from the point of view of manifold invariants?
The map $\pi_* tmf \to MF_*$ is a rational isomorphism, but not a surjection. As a result, there are certain values that the Witten genus does not take, just as the $\hat A$--genus of a Spin manifold of dimension congruent to 4 mod 8 must be an even integer (which implies Rokhlin's theorem). Some examples: $c_6$ is not in the image but $2c_6$ is, which forces the Witten genus of 12-dimensional String manifolds to have even integers in their power series expansion; similarly $\Delta$ is not in the image, but $24\Delta$ and $\Delta^{24}$ both are. (The full image takes more work to describe.)
The map $\pi_* tmf \to MF_*$ is also not an injection; there are many torsion classes and classes in odd degrees which are annihilated. These actually provide bordism invariants of String manifolds that aren't actually detected by the Witten genus, but are morally connected in some sense because they can be described cohomologically via universal congruences of elliptic genera. For example, the framed manifolds $S^1$ and $S^3$ are detected, and Mike Hopkins' ICM address that Drew linked to describes how a really surprising range of framed manifolds is detected perfectly by $\pi_* tmf$.
These results could be regarded as "the next version" of the same story for the relationship between the $\hat A$-genus and the Atiyah-Bott-Shapiro orientation for Spin manifolds. They suggest further stages. And the existence, the tools for construction, and the perspective they bring into the subject have been highly influential within homotopy theory, for entirely different reasons.
Hope this provides at least a little motivation.